Differentiable Function is Continuous/Corollary

Corollary to Differentiable Function is Continuous
Let $f$ be a real function defined on an interval $I$. If $f$ is not continuous at $x_0$, $f$ is not differentiable at $x_0$.

Proof
From Differentiable Function is Continuous, if $f$ is differentiable at $x_0$, then it is continuous at $x_0$.

By the Rule of Transposition, it follows that if $f$ is not continuous at $x_0$, then it can not be differentiable at $x_0$.