Determine if the Heaviside function $u_{0}( t) =\left\{ \begin{array}{l@{\quad}ll} 0 & \hbox{if} & t < 0 \\ 1 & \hbox{if} & t ≥ 0 \end{array} \right.$ has a derivative at $0.$

Solution Since $u_{0}( t)$ is discontinuous at 0, it has no derivative at 0.