[Math] Continuous functions out of a Product Space

general-topologysoft-question

Munkres states that there is no useful criterion for continuity if the domain of a function is a product space and gives an example:
$$F(x \times y) = \frac{xy}{x^2+y^2} \quad x \times y\neq 0 \times 0$$
And $F(0 \times 0) = 0$.
This is continuous in each variable separately $F(x \times y_0)$ and $F(x_0 \times y)$. But it fails to be continuous in the usual epsilon-delta sense at the origin.

Later though, we are asked to prove the metric $d(x,y)$ is continuous. What works in the case of the metric that doesn't work for $F(x \times y)$?

This is related to a previous question I asked: Clarification on the Continuity of Product Maps

Mainly it's all the same confusion — when there's a product in the domain how do I start investigating continuity? Evidently it makes sense sometimes (i.e. with the metric)

Best Answer

Saying that there is "no useful criterion for continuity" is a quite the overstatement. There is no "universal criterion", sure, and we must be careful, but there are some rules which do allow us to tell if some particular function is continuous. It's not like we don't even know if $f(x,y)=x$ is continuous or not (It is).

In the particular case of the metric being continuous, for example, there is an important fact that is left out, and that is which definition of continuity we are working in. There are two that are of importance here:

In metric spaces:

If $(X,d_X)$ and $(Y,d_Y)$ are metric spaces and $f:X\to Y$, then $f$ is continuous at $x_0\in X$ if, for every $\epsilon > 0$, there exists some $\delta>0$ such that if $d_Y(x,x_0)<\epsilon$, then $d_Y(f(x),f(x_0))<\epsilon$. In other words, $$\forall \epsilon >0\exists \delta>0: d_Y(x,x_0)<\epsilon\implies d_Y(f(x),f(x_0))<\epsilon$$ $f$ is continuous if it is continuous at all points in $X$.

In topological spaces:

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and $f:X\to Y$, then $f$ is continuous if the preimage of every open set in $Y$ is open in $X$. In other words: $$\forall O\in \tau_Y: f^{-1}(O)\in \tau_X$$

What the question you post is asking us is to:

  1. Take a metric space $(X, d)$
  2. Define a topology induced by the metric.
  3. use the topological definition of continuity and prove that under this definition, $d$ is continuous.
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