Definition:Differentiable Mapping/Real Function/Real Number Line

Definition
Let $f$ be a real function defined on $\R$.

By definition, $\R$ is an (unbounded) open interval.

Let $f$ be differentiable on the open interval $\R$.

That is, let $f$ be differentiable at every point of $\R$.

Then $f$ is differentiable everywhere (on $\R$).