If f is continous and exist real numbers show that $f(x)=0$ If $f$ is continous in $[a,b] \subset \mathbb{R}$ and exist real numbers $\alpha \not= \beta$ such that
$$\alpha\int_{a}^{c} f(x) dx + \beta\int_{c}^{b} f(x) dx  =0$$
for all $c \in [a,b]$ show that $f (x)=0$ for all $x \in [a,b].$

Let $g(c)=\alpha\int\limits_a^c f(x) dx + \beta \int\limits_c^b f(x) dx$, $g$ is derivable and since $f$ is continuous,  $g'(c)=(\alpha-\beta)f(c)=0$ since $g$ is constant. Since $\alpha\neq \beta$, we deduce that $f(c)=0, c\in [a,b]$.
