Suppose that $$ f_n$$ is a sequence of continuously differentiable functions on $$ [a,b]$$ such that $$ f_n^{\mathrm{\prime <!--\; \prime \; -->}}$$ converges uniformly to a function $$ g$$ and there is a point $$ c\in <!--\; \in \; -->[a,b]$$ such that $$ \underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{f}_n\left(c\right)=\mathrm{\gamma <!--\; \gamma \; -->}$$ exists. Then $$ f_n$$ converges uniformly to a differentiable function $$ f$$ with $$ f\left(c\right)=\mathrm{\gamma <!--\; \gamma \; -->}$$ and $$ f^{\mathrm{\prime <!--\; \prime \; -->}}=g$$