Suppose that $f : [a,b] \rightarrow R$ is continuous. Suppose that$\int\_a^x = \int\_x^b f $ for all $x ∈ [a,b]$. Show that $f(x)=0$ for all $x$ in $[a,b]$  

I have the following:  

$$\int\_a^xf- \int\_x^bf =0$$  

$$\implies\int\_a^bf=F(b)-F(a)=0$$  

$$\implies F(b)=F(a)$$.  

I am unsure if this is the right track, and if it is how to come to the conclusion.