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I always thought that strictly increasing functions always have inverses. However, now I am given a task to find a strictly increasing function $f(x)$ that is bounded between [0,1] and doesn't have an inverse $f(x)^{-1} : [f(0), f(1)] \rightarrow [0,1]$

I can't seem to come up with any examples that satisfy the criteria. Even discontinuous functions seem to have inverses. Any tips on this problem?

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    Maybe try making a discontinuous function that is not surjective onto $[f(0),f(1)]$. – 2017-02-23

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You seek a strictly increasing function $f\colon [0,1]\to [f(0),f(1)]$ that does not have an inverse. If $x

This leads us to the following counterexample: $$ f(x)=\begin{cases} \frac{x}{2},& 0\leq x\leq \frac{1}{2}\\ \frac{x+1}{2},&\frac{1}{2}