Fixed Point Theorem for Continuous Functions on a Closed Interval

Consider the function $g(x)=f(x)−x$. Think about the values of $g(x)$ at the endpoints $x=0$ and $x=1$, and use the Intermediate Value Theorem. What can you say about $g(x)$ on the interval $[0,1]$?

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Boundedness of Continuous and Periodic Functions on $\mathbb{R}$