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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?
• 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.
• What is epsilon-delta rigorous definition of limits?
• 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-

In these videos, Sal builds the definition step by step.
• How many methods are there to know whether a function is continuous or not?
• 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.
• Would f(x) = sqrt(x) be continuous? It starts at (0, 0) but its domain isn't all real numbers.
• 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.
• Can we call a function continuous if its limit at one point does not exist but the limit exists at the other point ?
• 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 → R` is a function defined on some non-empty subset `A` of `R`. Let `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`.
• What is the difference between a function f(x) being discontinuous, and a function f(x) not being differentiable?
• 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.
• 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?
• 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?
• 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
• 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.