Let $f(x)=(x-a)g(x)$. Calculate $f '(a)= g(a)$. Assuming that $g$ is continuos. What is $f(a)$ in this case? Can you give an explanation for this proof? How can it be possible to ask about a function of another variable of an existing function? Can you give other examples? This made me question my understanding of functions.

Another solution might be that if $f(a)=0$, then forming the difference quotient makes $$\frac{f(a+\Delta a)-f(a)}{\Delta a}=\frac{(((a+\Delta a)-a)g(x))-0}{\Delta a}=\frac{\Delta a \ g(x)}{\Delta a}=g(x).$$
(Hello to anybody running across this from OCW.)
