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## Class 12 math (India)

### Unit 5: Lesson 1

Continuity at a point# Continuity introduction

Sal introduces a formal definition of continuity at a point using limits. Created by Sal Khan.

## Want to join the conversation?

- Okay, here's an odd case: What about 1/x? 1/x is not defined at 0, but the limit of 1/x as x -> 0 is ALSO not defined. Or, rather, it doesn't exist. Does this mean that 1/x qualifies as continuous, or are "function is not defined" and "limit does not exist" considered different things? My intuition says "1/x is not continuous" simply because, well, just look at it.

Is there ANY function that's undefined at a given point that's still considered continuous, or must a function be defined at every point in order to be continuous?(31 votes)- If a function is undefined at any point, than it is NOT continuous. Being undefined at a point creates a hole in the function, therefore rendering it not continuous.(39 votes)

- What is epsilon-delta rigorous definition of limits?(29 votes)
- The delta-epsilon definition is a formal definition for limits. When you start to learn calculus, you usually figure out the limits from a look at a graph or by intuition; This definition is one of the strongest concepts of Calculus, or even at math entirely.

The formal definition is- f(x) as x approaches c is L, if for every d>0, there exists an e>0, such that if |x0-c|<e, then |f(x0)-L|<d.

It seems extremely complicated at first glance, but I suggest you to take a look at these videos-

https://www.khanacademy.org/math/calculus/limits_topic/epsilon_delta/v/limit-intuition-review

In these videos, Sal builds the definition step by step.(27 votes)

- How many methods are there to know whether a function is continuous or not?(11 votes)
- 1) Use the definition of continuity based on limits as described in the video:

The function f(x) is continuous on the closed interval [a,b] if:

a) f(x) exists for all values in (a,b), and

b) Two-sided limit of f(x) as x -> c equals f(c) for any c in open interval (a,b), and

c) The right handed limit of f(x) as x -> a+ equals f(a) , and

d) The left handed limit of f(x) as x -> b- equals f(b).

2) Use the pencil test: a continuous function can be traced over its domain without lifting the pencil off the paper.

3) A continuous function does not have gaps, jumps, or vertical asymptotes.

4) Differentiability implies continuity.

5) Classification of functions based on continuity. Examples:

All polynomial functions are continuous over their domain.

All rational functions are continuous except where the denominator is zero.

The composition of two continuous functions is continuous.

The inverse of a continuous function is continuous.

Sine, cosine, and absolute value functions are continuous.

Greatest integer function (f(x) = [x]) and f(x) = 1/x are not continuous.

Sign function and sin(x)/x are not continuous over their entire domain.(25 votes)

- Would f(x) = sqrt(x) be continuous? It starts at (0, 0) but its domain isn't all real numbers.(7 votes)
- A function is considered continuous if it meets the requirement for all numbers in its domain. It does not have to be continuous for all real numbers.(19 votes)

- Can we call a function continuous if its limit at one point does not exist but the limit exists at the other point ?(4 votes)
- You should be more clear about what you mean by
*"(…) but the limit exists at the other point?"*.

Recall that we call a function continuous on a given set if and only if it is continuous at every point of that set. As such, if things go bad at one point, it does not really help if things are well-behaved at some other point. There are three cases if the limit at a given point in the domain of the function fails to exist. They come down to the characterisation of the point in question.

To be rather precise, suppose`ƒ: A →`

is a function defined on some non-empty subset**R**`A`

of

. Let**R**`a ∈ A`

be a point where the limit`lim (x→a) ƒ(x)`

fails to exist.`Case I`

:`a`

is an*isolated point*of`A`

. We call`a`

an isolated point of`A`

if and only if there exists some neighbourhood of`a`

whose intersection with`A \ {a}`

is empty. This is the most trivial case. If`a`

is an isolated point of`A`

, it is not meaningful to speak of the limit of`ƒ`

at`a`

(since it is undefined). Hence, the limit does not exist. In spite of this,`ƒ`

is continuous at`a`

.`Case II`

:`a`

is an*interior point*of`A`

. This means that there is some neighbourhood of`a`

entirely contained in`A`

. Most texts on elementary calculus only define limits at such points. If the limit of`ƒ`

does not exist at such a point`a`

, then`ƒ`

is*not*continuous at`a`

, i.e., is discontinuous at`a`

.`Case III`

:`a`

is a*limit point*of`A`

, but not an interior point of`A`

. We call`a`

a limit point of`A`

if and only if every neighbourhood of`a`

contains some point of`A`

different from`a`

itself. Since`a`

is not an interior point of`A`

, the limit of`ƒ`

at`a`

does not exist (is undefined). In this case we can not assert any general conclusion that will hold for every such function`ƒ`

and point`a`

;`ƒ`

might be continuous at`a`

, or it might be discontinuous. If there exists a sequence`{a(n)}`

of points in`A`

such that`a(n) ≠ a`

for every`n`

and such that`a(n) → a`

(that is, the sequence converges to`a`

), but the sequence`{ƒ[a(n)]}`

does not converge to`ƒ(a)`

, then`ƒ`

is not continuous at`a`

. If no such sequence exists, then`ƒ`

is continuous at`a`

.(5 votes)

- What is the difference between a function f(x) being discontinuous, and a function f(x) not being differentiable?(5 votes)
- A function is not differentiable if it contains a point that has one instantaneous slope if measured from the positive x direction and a different slope if measured from the negative x direction. This is true even if the function is continuous.

You can sometimes restrict the domain and differentiate only a portion of the function that is differentiable, but the function as a whole would not be differentiable.(2 votes)

- Is the function
*f(x)= sqrt(x)*continuous? I understand that it is continuous for every value in its domain, but do you consider that it has no values for x<0?(2 votes)- No, that doesn't make it discontinuous.

I wasn't sure how to word this so wikipedia tells us:

"The function f is said to be continuous if it is continuous at every point of its domain; otherwise, it is discontinuous."

https://en.wikipedia.org/wiki/Continuous_function#Real-valued_continuous_functions

And yes, it is continuous.(3 votes)

- If a function contains an absolute value of some sort, can it then be continuous over some interval even if it's not derivable all over that interval?(4 votes)
- Yes it can. In fact absolute values are "smoother" in a sense.(1 vote)

- How is the Epsilon-Delta definition used or even needed for Calculus, both differentiation and integration, I know that they are used but I don't understand why or how, please and thank you(3 votes)
- No.

Answer is we do not even know what a limit is without the eps-delta definition.

Without the definition "lim" is just some symbol without meaning.(2 votes)

- So I'm guessing a graph is not defined as continuous if you have a situation where one side - or both - of a limit is approaching infinity (hence the limit does not exist, as has been discussed), but I was wondering what exactly this discontinuity is called, if it has a name.(3 votes)
- There are many kinds of discontinuity, some of them are:

1) Missing point - Picture a smooth curve with just one "hole" in it, that is the function is not defined at that point

2) Isolated point - Similar to case 1), except the function is defined at that point, just not in the line of the curve

3) Jump - Eg: The function consists of two smooth curves which aren't connected, like the first example Sal shows

4) Infinte - When the graph goes to positive or negative infinity and comes back from positive or negative infinity, think tanx graph

Hope this helped!(2 votes)

## Video transcript

What I want to do in this
video is talk about continuity. And continuity of a
function is something that is pretty easy to
recognize when you see it. But we'll also talk about how we
can more rigorously define it. So when I talk about it being
pretty easy to recognize, let me draw some functions here. So let's say this is the
y-axis, that is the x-axis. And if I were to
draw a function, let's say f of x looks
like something like this. And I would say
over the interval that I've drawn it, so
it looks like from x is equal to 0, to maybe
that point right over there, is this function continuous? Well you'd say no, it isn't. Look, we're over here,
we see the function just jumps all of a sudden, from
this point to this point right over here. This is not continuous. And you might even say
we have a discontinuity at this value of
x, right over here. We would call this
a discontinuity. And actually this
type of discontinuity is called a jump discontinuity. So you would say this
is not continuous. It's obvious that these
two things do not connect. They don't touch each other. Similarly, if you were to
look at a function that looked like-- let me draw
another one-- y and x. And let's say the function
looks something like this. Maybe right over here,
looks like this, and then the function is defined to be
this point right over there. Is the function continuous
over the interval that I've depicted
right over here? And you would immediately
say no, it isn't. Because right over
at this point, the function goes up to
this point, just like this. And this kind of discontinuity--
this is the discontinuity-- is called a removable
discontinuity. Removable. One could make a reasonable
argument that this also looks like a jump,
but this is typically categorized as a
removable discontinuity. Because if you just
re-define the function so it wasn't up here, but
it was right over here, then the function is continuous. So you can kind of
remove the discontinuity. And then finally, if I were
to draw another function-- so let me draw another one
right over here-- x, y. And ask you, is this one
continuous over the interval that I've depicted? And you would say, well, look. Yeah, it looks all
connected all the way. There aren't any
jumps over here. No removable
discontinuities over here. This one looks continuous. And you would be right. So that's the general
sense of continuity. And you can kind of
spot it when you see it. But let's think about a more
rigorous definition of one. And since we already have a
rigorous definition of limits, the epsilon delta
definition, gives us a rigorous definition
for limits. It's a definition for limits. So we can prove
when a limit exists, and what the value
of that limit is. Let's use that to create
a rigorous definition of continuity. So let's think about a function
over some type of an interval. So let's say that we have-- so
let me draw another function. Let me draw some
type of a function. And then we'll see whether
our more rigorous definition of continuity
passes muster, when we look at all of
these things up here. So let me draw an
interval right over here. So it's between that x
value and that x value. This is the x-axis,
this is the y-axis. And let me draw my function
over that interval. Over that interval it
looks something like this. So we say that a
function is continuous at an interior point. So an interior point
is a point that's not at the edge of my boundary. So this is an interior
point for my interval. This would be an end
point, and this would also be an end point. We'd say it's continuous
at an interior point. So continuous at interior
point, interior to my interval, means that the limit as,
let's say at interior point c, so this is the point
x is equal to c. We can say that it's continuous
at the interior point c if the limit of
our function-- this is our function
right over here-- if the limit of our
function as x approaches c is equal to the value
of our function. Now does this make sense? Well, what we're saying
is, is at that point, well this is f of
c right over there. And the limit as
we approach that is the same thing as the
value of the function. Which makes a lot of sense. Now let's think about it. If these would have somehow
been able to pass for continuous in that context. Well, over here, let's say that
this is our point c. f of c is right over there. That is f of c. Now is it the case
that the limit of f of x, as x approaches
c, is equal to f of c? Well, if we take
the limit of f of x as x approaches c from
the positive direction, it does look like it is f of c. It does look like
it's equal to f of c. But if we take the
limit-- but this does not equal the
limit of f of x as x approaches c from
the negative direction. As we go from the
negative direction, we're not approaching f of c. So therefore, this
does not hold up. In order for the limit
to be equal to f of c, the limit from
both the directions needs to be equal to it. And this is not the case. So this would not pass muster
by our formal definition, which is good. Because we see visually
this one is not continuous. What about this one
right over here? And let me re-set it up. So let me make sure
that that looks like a hole right over there. So we see here,
what is the limit? The limit-- and
this is our c, right over here-- the limit of
f of x as x approaches c, let's say that
that is equal to L. And so that, we've seen many
limits like this before, that's L right over there. And it's pretty
clear just looking at this is that L
does not equal f of c. This right over here is f of c. So once again, this
would not pass our test. The limit of f of
x as x approaches c, which is this right over
here, is not equal to f of c. So once again, this
would not pass our test. And here, any of the interior
points would pass our test. The limit as x
approaches this value is equal to the function
evaluated at that point. So it seems to be
good for all of those. Now let's give a
definition for when we're talking about
boundary points. So this is continuity
for an interior point. And let's think about
continuity at boundary-- or let me call it
endpoint, actually, that would be better-- at endpoint c. So let's first consider
just the left endpoint. If left endpoint-- so what I'm
talking about, a left endpoint? Let me draw my axes,
x-axis, y-axis. And let me draw my interval. So let's say this is the
left endpoint of my interval, this is the right
endpoint of my interval. And let me draw the
function over that interval. Looks something like this. So when we talk about
a left endpoint, we're talking our c
being right over here. It is the left endpoint. So if we're talking
about a left endpoint, we are continuous at
c means-- or to say that we're continuous at
this left endpoint c-- that means that the limit f
of x as x approaches c, well, we can't even approach see
from the left hand side. We have to approach
from the right. Is equal to f of c. And so this is
really kind of a, we can only approach something
from one direction. So we can't just say
the limit in general, but we can say the
limit from one side. So it's really very
similar to what we just said for an interior point. And we see over here
it is indeed the case, as x approaches c, our
function is approaching this point right over here. Which is the exact
same thing as f of c. So we are continuous
at that point. What's an example where an
endpoint-- where we would not be continuous and an end point? Well, I can imagine a graph
that looks something like this. So here's our interval,
and maybe our function. So at c it looks like that. There's a little
hole right there. And then it would look
something like that. Or there's no hole,
the function just has a removable discontinuity
right over there. At least visually
it looks like that. And you see that this
would not pass the test. Because the limit as we approach
c from the positive direction is right over here. That's the limit. But f of c is up here. So f of c does not
equal the limit as x approaches c from
the positive direction. So this would not be continuous. And you could
imagine, what do we do if we're dealing
with the right endpoint? So we say we're continuous
at right endpoint c if-- so let me draw that, do
my best attempt to draw it-- so this is my x-axis,
this is my y-axis. Let me draw my interval
that I care about, say it looks
something like this. A right endpoint means
c is right over there. And we can say that we
are continuous at x, the function is
continuous at x equals c means that the limit of f of
x as x approaches c-- now we can't approach it from both sides. We can only approach it
from the left hand side. As x approaches c from
the negative direction, is equal to f of c. If we can say that,
if this is true, then this implies that we
are continuous at that right endpoint, c, and vice versa. And a situation where we're not? Well you could imagine instead
of this being defined right at that point, you
could create, you could say the function jumps up. Just like we did
right over there. So once again, continuity, not
a really hard to fathom idea. Whenever you see the function
just all of a sudden jumping, or there's kind of a gap in
it, it's a pretty good sense that the function is
not connected there. It's not continuous. But what we did
in this video is, we used limits to define
a more rigorous definition of continuity.