Is a strictly decreasing, invertible function necessarily continuous? We know that the inverse exists (therefore injective and surjective), and it is strictly decreasing. I'm not sure if continuity follows.
I don't think it necessarily follows. For example the split function:

$f: [-3;3] \rightarrow [-3;-1]\cup[0;4]$,  

$f(x)= \mathrm{if}(x&lt;1, -x+1, -x)$.

It is discontinuous at $x=1$, but it is still invertible and strictly decreasing.
I don't think it necessarily follows. For example the split function:

$f: [-3;3] \rightarrow [-3;-1]\cup[0;4]$,  

$f(x)= \mathrm{if}(x&lt;1, -x+1, -x)$.

It is discontinuous at $x=1$, but it is still invertible and strictly decreasing.