Differentiation of subtraction I've got an exercise to do and I don't really know what to do.

Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist $(f-g)'(a)$? Explain it. 

My opinion is that exists, but I've got no idea how should I explain it. Some help? Thank you.
$f(a)=0$ and $\exists f'(a)$.

The expression : $(f-g)'(a)$ is : $f'(a) - g'(a)$.

You cannot say that this derivative exists, if $g$ is not differentiable at $a$. In your question's body, you have only stated that $f$ is differentiable and that $g$ is only continuous. Thus, no, you cannot say generally that $\exists (f-g)'(a)$.
Is $g$ differentiable?  That information is not given.  It is possible for $g$ to be continous but not differentiable.

Anyway differentiation is "linear", in that $(f-g)'(a)$ = $f'(a) - g'(a).$
So, if $g'(a)$ exists (and $f'(a)$ exists which is given) then $(f-g)'(a)$ exists.

But if $g'(a)$ does not exist, then $(f-g)'(a)$ does not exist.