Show that if the integral of a function from 0 to x is equal to the integral from x to 1, then f(x)=0

Question

How can I prove that if $f$ is continuous on $[0, 1]$ such that for every $x \in [0, 1]$ we have that $\int_{0}^{x} f(t) \,dt = \int_{x}^{1} f(t) \,dt$, then $f(x)=0$ for every $x$.

Answer 1

$$2\int_0^x f(x)\,dx=\int_0^x f(x)\,dx+\int_x^1 f(x)\,dx=\int_0^1 f(x)\,dx=\text{Cst}.$$