Question #5259

give an example to show that evert continuos function is not necessarily differentiabe

Expert's answer

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.

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.

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