calculus - Prove existence of a point in continuous function

Let $f(x)$ be a continuous function in $\left [ 0,1 \right ]$ so that for every $x\in \left [ 0,1 \right ]$ there is $f(x)\in \left [ 0,1 \right ]$.

Prove that there such a point $x_0\in \left [ 0,1 \right ]$ so that $x_0 + f(x_0) = 1$

Could you please point me to the right direction with this question?
I Tried proving it using Stone–Weierstrass theorem and intermediate value theorem but didn't got anywhere.

Thanks!

Answer

The function $g(x) := x + f(x)$ is continuous on $[0,1]$ and has $g(0) = f(0) \le 1$ and $g(1) = 1 + f(1) \ge 1$. Hence, by the intermediate value theorem, $g(x_0) = 1$ for some $x_0 \in [0,1]$, i.e., $x_0 + f(x_0) = 1$ for some $x_0 \in [0,1]$.