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### Course: AP®︎/College Calculus AB>Unit 1

Lesson 11: Defining continuity at a point

# Continuity at a point

Saying a function f is continuous when x=c is the same as saying that the function's two-side limit at x=c exists and is equal to f(c).

## Want to join the conversation?

• This video should be shown from the very beginning of this Unit.
• Just completed the section named Continuity at a Point (3/27/20 at ).

Sal employed a condition in this section (and several previous sections) that says "if and only if." I know that there are two parts to this type of condition: the "if" part and the "only if" part, but I'm not quite sure why they are different. Can anyone explain that to me?

Regards- Charlie Reynolds
• If A and B are statements, then 'A if and only if B' means 'A and B are logically equivalent.' If either statement is true, so is the other. If either statement is false, so is the other. This is called a biconditional statement, and it often gets abbreviated as 'A iff B.'

We can break the biconditional into two parts: 'A if B' and 'A only if B'.

'A if B' can be rewritten as 'if B, then A' since the 'if' is being applied to statement B.

'A only if B' means 'if not B, then not A'. This is the effect of the word 'only'.

Finally, 'if not B, then not A' is equivalent to 'if A, then B.' These two statements are called the contrapositives of one another.

So taken together, 'A iff B' means 'if A then B and if B then A'.
• For the third example, at , I understand that the graph is not continuous as x approaches c; however, would the graph be continuous if you specifically said as x approaches c from the left? That statement satisfies the definition relayed at the beginning of the video
• The graph would still not be continuous as a whole. BUT, the CLOSED INTERVAL from negative infinity to c would be continuous, which is what the definition of continuity implies.
• Why is is that the content order for this unit seems to jump around a lot in terms of the order that it should be taught? It seems like this should be one of the first lessons as it defines the definition of continuity of a point, something we used in previous lessons like solving limits.
• At the teacher introduce f(c) in a place different from the function...my question is what happens if this point doesn't exists. What if c is not part of the domain?
• If c is not part of the domain, the graph would have a hole at c, and f(c) would not exist. It would then be impossible for the limit as x approaches c to equal f(c), so the function is still not continuous.
• I know this is a silly question; more of a joke honestly. And to clarify, I know the answer. But if the formal definition of whether a function is continuous is lim_x->c f(c) = f(c), and you have a graph with a jump discontinuity at both ends of a point...

Example f(x)={x if 0 < x < 2, 5 - x if 2 < x < 4}

Since both the limit and f(x) are undefined at x = 2, would the formal definition be proving the graph continuous??
• No. Non-existence is not a value that a limit or a function can take on. The limit must exist in order to be equal to something else, and same with the function value.
• If the function is defined over a closed interval, how will we determine continuity at the endpoints? The two-sided limits don't exist for the endpoints.
• If a function f is only defined over a closed interval [c,d] then we say the function is continuous at c if limit(x->c+, f(x)) = f(c). Similarly, we say the function f is continuous at d if limit(x->d-, f(x))= f(d).

As a post-script, the function f is not differentiable at c and d.
• So in the limit notation, when x approaches to the given value , the y value would always be Undefined?
(1 vote)
• Not always. The y value is only undefined if there is a discontinuity/hole there.