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For the function to be differentiable, its derivative has to exist everywhere, which means the derivative itself must be continuous. Differentiating gives
The question mark is a placeholder, and if the derivative is to be continuous, then the question mark will have the same value as the limit as
from either side.
So the derivative will be continuous as long as
For the function to be differentiable everywhere, we need to require that
is itself continuous, which means the following limits should be the same:
So, the function should be
with derivative
The question mark is a placeholder, and if the derivative is to be continuous, then the question mark will have the same value as the limit as
So the derivative will be continuous as long as
For the function to be differentiable everywhere, we need to require that
So, the function should be
with derivative