The derivative of the given example function

*does* exist. Remember that the derivative of a function is given by

.

Given the function

we calculate the derivative at

by computing

.

Compute this and see what you find. Then compute (using standard differentiation techniques) the

for

. Note that

is continuous if and only if

exists and is equal to

. So does the limit exist ?

EDIT: However I would be interested in knowing if it's possible to find an example of a function

which is differentiable on some domain

such that left- and right-sided limits of

exist on all of

, but such that

is discontinuous in the particular sense of having the left- and right-sided limits unequal at some

. Perhaps that is impossible.