Verify that $\varphi$ is of class $\mathscr{C}^{0}$ on $[0, +\infty[$ and $\mathscr{C}^{\infty}$ on $]0, +\infty[$. Give the limit of the derivative $\varphi'(t)$ of $\varphi$ as $t$ tends to 0 in $]0, +\infty[$.
Where $\varphi$ is defined by
$$\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$$