Give an example of a function fRR which is differentiable at

Give an example of a function f:R--->R which is differentiable at only one point, and prove that your example is correct. (i.e. prove that that f is differentiable at that unique point and that it is not differentiable at any other point.)

Solution

Consider the function

f(x) = x, if x is rational

= -x, if x is irrational

f is well defined

f\'(x) = 1 if x is rational and -1 if x is irrational

As around every rational number there will be infinite irrational and vice versa it follows that

f is not differentiable at any rational or irrational value.

The exception is the point 0 where f(x) =0

and f is differentiable at x =0 with derivative = 0