Topological proof that a continuous injective real function on an interval has a continuous inverse

The intermediate value theorem is all that you need to show that $f$ is either strictly increasing or strictly decreasing, and that $f[A]$ is order-convex, meaning that whenever $x<y<z$, and $x,zin f[A]$, then $yin f[A]$. This means that $f[A]$ must be all of $Bbb R$, a ray, or an interval, i.e., that it must have one of the following forms: $Bbb R$; $(leftarrow,a)$, $(leftarrow,a]$, $(a,to)$, or $[a,to)$ for some $ainBbb R$; or $(a,b)$, $[a,b)$, $(a,b]$, or $[a,b]$ for some $a,binBbb R$. The intermediate value theorem is also all that you need to show that if $[a,b]subseteq A$, then

$$fbig[[a,b]big]=begin{cases} [f(a),f(b)],&text{if }ftext{ is increasing}\ [f(b),f(a)],&text{if }ftext{ is decreasing.} end{cases}$$

Now suppose that $(a,b)subseteq A$. Note that $a$ and $b$ need not belong to $A$. However,

$$(a,b)=bigcup_{a<c<d<b}[c,d];,$$

so

$$begin{align*} f[(a,b)]&=bigcup_{a<c<d<b}fbig[[c,d]big]\ &=begin{cases} left(inf_limits{a<c<b}f(c),sup_limits{a<c<b}f(d)right),text{ if }ftext{ is increasing}\ left(inf_limits{a<c<b}f(c),sup_limits{a<c<b}f(c)right),text{ if }ftext{ is decreasing};. end{cases} end{align*}$$

(It’s possible that $inf_limits{a<c<b}f(c)=-infty$ or $sup_limits{a<c<b}f(c)=+infty$; in that case $fbig[(a,b)big]$ is $Bbb R$ or an open ray.)

If $a=min A$, a basic open nbhd of $a$ has the form $[a,b)$ for some $a<bin A$, and

$$fbig[[a,b)big]=begin{cases} big[f(a),f(b)big),&text{if }ftext{ is increasing}\ big(f(b),f(a)big],&text{if }ftext{ is decreasing.} end{cases}$$

A corresponding result holds for intervals of the form $(a,b]$ if $b=max A$, so $f$ is open as well as bijective, and the inverse of an open bijection is continuous.

If the function $f$ is injective, and continuous then it is monotone, by the intermediate value theorem. Then you need only note that the open sets on both $A$ and $f(A)$ are induced by their orderings. Note that $f(A)$ is also an interval, again by the intermediate value theorem.

Note this works for open intervals $A$ as well as compact ones.

Snake eating its tail (mentioned in comments). Red arcs indicate where open ends of interval go.

Use the following theorem from topology:

If $f:X rightarrow Y$ is a continuous bijection of topological spaces where $X$ is compact and $Y$ is Hausdorff then $f$ is a homeomorphism (has continuous inverse).

https://proofwiki.org/wiki/Continuous_Bijection_from_Compact_to_Hausdorff_is_Homeomorphism