Let $\displaystyle S_n(x)= \displaystyle \frac{1}{n}e^{-n^2x^2}$. Show that there is a function, $\displaystyle S(x)$, such that $\displaystyle S_n(x)-->S(x)$ uniformly on R and that $\displaystyle S'_n(x)-->S'(x)$ for every x but that the convergence of the derivatives is not uniform in any interval which contains the orgin.