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Specify a bijective function $f:[0,1] \longrightarrow [0,1)$

I know that such a function can't be continuous, so I've tried to find a function with one or more discontinuities, but with no progress so far.

Tom
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    I am sure I have seen this question on this site before, but in the meantime. Consider the sets $A={1,1/2,1/3,\dots}$ and $B={1/2,1/3,1/4,\dots}$ can you find a bijection between them? Can you then extend this bijection to have domain $[0,1]$ and range $[0,1)$. –  Nov 23 '20 at 20:49

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Notice that it's fairly east to find a bijection between two countable infinite set. Now simply choose any countable infinite subset of $[0,1)$, put the extra end $1$ to $A$ to get a new set $B$, and you can have a bijective map from $A$ to $B$. Keep other elements unchanged and you just found a bijection form $[0,1)$ to $[0,1]$.

Ezrel Cris
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