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Answer 1 · 2016-02-18T18:31:51

It is false.

For instance $|x|$ is continuous, but does have derivate in $x=0$ .

Additionally to the continuity, if have to guarantee that $f'(c^-)=f'(c^+)$

Answer 2 · 2016-02-19T01:10:06

Another example is $f(x) = root(3)x$

$f(x) = root(3)x$ is continuous at $0$ , but $f'(x) = 1/(3root(3)(x^2))$ is not defined at $0$ .

It is possible to describe a function that is continuous everywhere, but differentiable nowhere. One example is the Weierstrass function.