Question

It's true that if f is differentiable in (a,b) then f is continuous in (a,b).

But would it be correct to say that if f is differentiable in (a,b) then f is continuous in [a,b] ?

Answer 1

I don't think it's necessarily true. I mean, it could be true, but could also be false.

Answer 2

All differentiable functions are continuous on that same interval.SO THE FIRST STATEMENT IS TRUE. but
NOT all continuous functions are differentiable .SO FOR THE SECOND STATEMENT I THINK IT IS FALSE.

Answer 3

Pasta is right. Just consider most rational power functions and the absolute value function.

Answer 4

If f(x) is differentiable at (a,b), then it has to be continuous the the same point.
But, it is not necessary that every continuous function is differentiable.

For example:
f(x) = |x|

Hence we conclude that
1) Differentiability implies continuity
2) Continuity does not imply differentiability.