# About Definition 4.5, Theorem 4.6, Theorem 4.8 in “Principles of Mathematical Analysis” by Walter Rudin.

Definition 4.5

Suppose $$X$$ and $$Y$$ are metric spaces, $$E \subset X, p \in E$$ and $$f$$ maps $$E$$ into $$Y$$. Then $$f$$ is said to be continuous at $$p$$ if for every $$\varepsilon > 0$$ there exists a $$\delta >0$$ such that $$d_Y(f(x),f(p)) < \varepsilon$$ for all points $$x \in E$$ for which $$d_X(x,p) < \delta$$.

Theorem 4.6

In the situation given in Definition 4.5, assume also that $$p$$ is a limit point of $$E$$. Then $$f$$ is continuous at $$p$$ if and only if $$\lim_{x \to p} f(x) = f(p)$$.

Rudin didn't write Definition 4.5 as follows:

Definition 4.5'

Suppose $$X$$ and $$Y$$ are metric spaces, $$p \in X$$ and $$f$$ maps $$X$$ into $$Y$$. Then $$f$$ is said to be continuous at $$p$$ if for every $$\varepsilon > 0$$ there exists a $$\delta >0$$ such that $$d_Y(f(x),f(p)) < \varepsilon$$ for all points $$x$$ for which $$d_X(x,p) < \delta$$.

And Rudin didn't write Theorem 4.6 as follows:

Theorem 4.6'

In the situation given in Definition 4.5', assume also that $$p$$ is a limit point of $$X$$. Then $$f$$ is continuous at $$p$$ if and only if $$\lim_{x \to p} f(x) = f(p)$$.

And Rudin wrote Theorem 4.8 as follows:

Theorem 4.8

A mapping $$f$$ of a metric space $$X$$ into a metric space $$Y$$ is continuous on $$X$$ if and only if $$f^{-1}(V)$$ is open in $$X$$ for every open set $$V$$ in $$Y$$.

Why?

Why is $$E$$ necessary in Definition 4.5, Theorem 4.6, Theorem 4.7?

I think $$E$$ is redundant.

• If you take $(E, d|_{E\times E})$ as the metric subspace, then you can see the two defintions are in fact the same. – lEm Jan 31 at 11:14
• Thank you, lEm for your comment. – tchappy ha Jan 31 at 13:23

You are right here. He can simply discard the complement of $$E$$ in $$X$$, and view it as a function from $$E$$ to $$Y$$. As IEm points out, E is a metric space in its own right using the metric inherited from $$X$$.
• This raises a meta-question: what was Rudin thinking by including $E$ and a remark that it's useless? He doesn't strike me as one to include fluff. – Solomonoff's Secret Jan 31 at 15:24