Let $X$ is be a compact Hausdorff topological space and let $d:C(X)\to C(X)$ is its algebra of be a not-necessarily continuous functionsderivation. Let $f\in C(X)$ and let $x\in X$. I want to show that $d(f)(x)=0$, and to do so it is enough to consider the case in which $f$ is real-valued and non-negative, then for every derivation element of $\delta:C(X)\to C(X)$ vanishesis a $\mathbb C$-linear combination of such functions. But in that case there exists a function $g\in C(X)$ such that $f=g^2$, and then $d(f)(x)=2g(x)d(g)(x)=0$ because $g(x)=0$.
(Note that by [Sakai, Shôichirô. On a conjecture of Kaplansky. Tôhoku Math. J. (2) 12 1960 31--33. MR0112055 (22 #2913)] a derivation of a $C^*$-algebra is automatically continuous, so imposing continuity in this context does not change much)