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# 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?

• 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 • 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?
(1 vote) • 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??
(1 vote) • 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.
(1 vote) • So in the limit notation, when x approaches to the given value , the y value would always be Undefined?  