regarding function continuous on closed interval

Question

f is said to be continuous in the closed interval [a, b] if

 f is continuous in (a, b)

 lim

x→a+ f (x) = f (a)

 lim

x→b- f (x) = f (b)

the limit at the end points should be defined from both the sides( i.e x→a- and x→b+) i know this is a closed interval but for a limit to exist,it has to be defined from both sides.similarly when we calculate derivative on closed interval this is the condition. why is it so?? can anyone explain?

Answer 1

There are terms like continuity from the right (a+) and continuity from the left (b-).

For that, only one-sided limit has to exist.

As for the derivative - continuity is not enough. For example, f(x) = |x| doesn't have a defrvative at x=0 (sharp point, not smooth) - and the derivative  limit doesn't exists

Answer 2

For continuity at the end points the limit only needs to be defined from the right side for a, and the left for b

For differentiability at the end points the limits at a and b need to exist from both sides, and the right and left limits need to be equal