Prove $T: X \to X$ has a fixed point if there exists an $x \in X$, so that $T(x) = x$

Question

Let $X$ be a non-empty set. A function $T: X \to X$ has a fixed point if there exists an $x \in X$, so that $T(x) = x$.

Let $f:[0,1] \to [0,1]$ be continuous.

How can one prove that $f$ has a fixed point? And can $f$ have two fixed points?

I thought of looking at the function $g:[0,1] \to \mathbb{R}$ given by $g(x) := f(x)-x$

If we take $a = 0$ and $b = 1$. We assume $f:[0,1] \to [0,1]$ has no fixed point. Then $[0,1] = \{x \in [0,1] : f(x) < x\} \cup \{x \in [0,1] : f(x) > x \}$. But how can we argue that this is not possible?

And I don't know if it's possible for $f$ to have two fixed points either.

Answer 1

Hint: Try to use Intermediate Value Theorem to the function $g$.