Answer to Question #5259 in Calculus for Akhtar Rasool khan
give an example to show that evert continuos function is not necessarily differentiabe
Let f(x) = |x|.
Then f(x) is continuous everywhere,
however it is not
differentiable at x=0.
f(x) = -x, x<0
Hence f is continuous on each of the intervals
0) and (0,+infitinity].
Moreover, at x=0 we have that
and therefore f is also continuous at 0.
On the other,
f'(x) = -1, x<0
= 1, x>0,
derivative of f is discontinuous at 0.