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What are some examples of functions which are continuous, but whose inverse is not continuous?

nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.

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What does "bicontinuous" mean? –  Chris Eagle Sep 30 '11 at 16:06
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@Chris: Probably that the inverse is continuous. –  Ted Sep 30 '11 at 16:10
    
Bicontinuous is a standard definition in some topology texts. See Gamelin and Greene –  JeremyKun Oct 2 '11 at 21:22
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6 Answers

up vote 14 down vote accepted

Define $f: [0,1) \cup [2,3] \rightarrow [0,2]$ by

$$f(x)=\begin{cases} x & x \in [0,1) \\ x-1 & x \in [2,3] \end{cases}$$

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A bijective map that is continuous but with non-continuous inverse is the following parametrization of the unit circle $\mathbb{S}^1$:

$$f: \colon [0, 2\pi) \to \mathbb{S}^1, \qquad f(\theta)=e^{i \theta}.$$

This map cannot have continuous inverse, because $\mathbb{S}^1$ is compact, while $[0, 2\pi)$ is not. Indeed, $f^{-1}$ jumps abruptly from $2\pi$ to $0$ when we travel round the unit circle.

Another example, somewhat similar in nature, is the map $g\colon [0,1] \cup (2, 3] \to [0, 2]$ defined by

$$g(x)=\begin{cases} x & 0 \le x \le 1 \\ x-1 & 2 < x \le 3 \end{cases}$$

The inverse map is $$g^{-1}(y)=\begin{cases} y & 0 \le y \le 1 \\ y+1 & 1 < y \le 2\end{cases}$$

and it is not continuous because of a jump at $y=1$. Note that, again, the range of $g$ is compact while the domain is not.

More generally, every bijective map $h\colon X \to K$ with $X$ non-compact and $K$ compact cannot have a continuous inverse.

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Let $X$ be a set and $\tau_1,\tau_2$ two topologies on $X$ with $\tau_2\subsetneq\tau_1$. Then the identity function from the topological space $(X,\tau_1)$ to $(X,\tau_2)$ is a continuous bijection but the inverse function (the identity function from $(X,\tau_2)$ to $(X,\tau_1)$) is not continuous.

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Let $\rm X$ be the set of rational numbers with the discrete topology. Then the identity map $\rm X\to \mathbb{Q} $ is bijective and continuous, with discontinuous inverse.

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If the inverse map also goes to $\mathbb{Q}$, then how is it discontinuous? –  isomorphismes Sep 30 '11 at 16:23
    
A bijective continuous map is a homeomorphism if and only if it is also an open map. A singleton is open in $X$, but not in $\mathbb{Q}$ (under the metric topology). –  Zhen Lin Sep 30 '11 at 17:34
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Take a "8"-shaped plane curve $\mathcal C \subset \mathbb R^2$ endowed with the subspace topology. Let $\phi: \mathbb R \to \mathcal C$ be a continuous injective parametrization of $\mathcal C$. The inverse function $\phi^{-1}: \mathcal C\to\mathbb R$ cannot be continuous because $\phi^{-1}((a,+\infty))$ is not an open set for some $a$.

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1) Take any topological space,
2) Obtain another space by refining its topology,
3) ...
4) PROFIT!

In fact, consider the forgetful functor $F: \mathbf{Top} \to \mathbf{Set}$. For any set $S$ the continuous functions of the form $f: X \to Y$ such that $FX = FY = S$ and $Ff = 1_S$ form a linear order on the set of all topologies on $S$, and this order is in fact inverse to the usual one (formed by set inclusion of topologies).

For example, extending the answer by Marco, consider a simple curve $\gamma: I \to M$ on some manifold with the finite number of self-intersections. For each intersection, remove all corresponding points from $I$ except for one. Voila :) UPD: actually, you can remove all corresponding points, period!

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