Putnam 1966 A5

A-5. Let $C$ denote the family of continuous functions on the real axis. Let $T$ be a mapping of $C$ into $C$ which has the following properties:

1. $T$ is linear, i.e. $T\left(c_{1} \psi_{1}+c_{2} \psi_{2}\right)=c_{1} T \psi_{1}+c_{2} T \psi_{2}$, for $c_{1}$ and $c_{2}$ real and $\psi_{1}$ and $\psi_{2}$ in $C$.
2. $T$ is local, i.e. if $\psi_{1} \equiv \psi_{2}$ in some interval $I$ then also $T \psi_{1} \equiv T \psi_{2}$ holds in $I$.

Show that $T$ must necessarily be of the form $T \psi(x)=f(x) \psi(x)$ where $f(x)$ is a suitable continuous function.