**Example:** Consider the function

This function looks like this:

We claim that it has no antiderivative on

Assume that some function *F* is an antiderivative of *f* on
*x* > 1*F* ′(*x*) = *f* (*x*) = 2,*x* < 1*F* ′(*x*) = *f* (*x*) = 1.

What can be said about the derivative at
*x* = 1?*F* is an antiderivative, we
must have
*F* ′(1) = *f* (1) = 2.*F* at 1 from the left must exist and
be equal to 2. However, since *F* is continuous at 1 from the left and
*F* ′

This contradiction shows that our assumption was wrong, *F* cannot be an
antiderivative of *f*.