Integrating a continuous function

Question

Let $a <b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be continuous functions. Suppose that for all $x \in [a,b]$ that $\int_a^x f(t) dt = \int_x^b f(t) dt$. Prove that $f(x) = 0$ for all $x \in [a,b]$

I know its basing off of The Fundamental Theorem of Calculus II. And I thought differentiating like the answer below was a way to grasp the concept but I just think I am over complicating it in my mind. I have a habit of doing that.

Answer 1

Hint. If you differentiate $$\int_a^x f(t) dt = \int_x^b f(t) dt$$ you obtain $$ f(x)=-f(x). $$