Question

The Honourable Expert pease send me an example to show that every continuos function is not necessarily differentiable.

Answer 1

Let f(x) = |x|.
Then f(x) is continuous everywhere,
however it is not
differentiable at x=0.

Indeed,

f(x) = -x, x<0
x,
x>=0

Hence f is continuous on each of the intervals
[-infinity,
0) and (0,+infitinity].

Moreover, at x=0 we have that
-0 =
0,
and therefore f is also continuous at 0.

On the other,
hand,

f'(x) = -1, x<0
= 1, x>0,

so the
derivative of f is discontinuous at 0.