Taken from an old exam, I hope I don't translate it unclear.
Let $f: \left [ 0,1 \right ] \rightarrow \mathbb{R}$ continuous and
let $f(\left [ 0,1 \right ]) \subset \left [ 0,1 \right ]$. By using
intermediate value theorem, show that $f$ has at least got one fixed
point in $\left[0,1\right]$ (aka there is one solution $x \in
\left[0,1\right]$ of the equation $f(x) = x$).
I'm not sure at all if I did it right but I have taken the function:
$$f(x) = x$$
Then take the given intervals and insert for $x$ the beginning of interval:
$$f(0)=0$$
And the end of the interval:
$$f(1) = 1 >0$$
Because the function is continuous and because the equality sign changes to an inequality sign right after, the intermediate value theorem provides there must be at least one solution $x_{1} = [0,1]$.
Did I do it correctly? Did I explain correctly?
Best Answer
Hint: Apply the intermediate value theorem to the function $g(x)=f(x)-x$. Note that $g(0) \geq 0 \geq g(1)$.