Limits and Continuity in Metric Spaces

11 Limits and Continuity in Metric Spaces

The concept of "limit" can be extended to metric spaces in a natural way, just by using distance in place of absolute value in the usual definitions. We can define both the limit of a sequence and the limit of a function in this way:

Definition: Let $(M,d)$ be a metric space, let $\{x_n\}_{n=1}^\infty$ be a sequence of points in $M,$ and let $L\in M,$ We say $\ds\lim_{n\to\infty}x_n = L$ if for all $\eps>0,$ there is an $N\in\N$ such that for all $n>N,$ $d(x_n,L)<\eps.$

Definition: Let $(A,\rho)$ and $(B,\tau)$ be metric spaces, let $f: A\to B$ be a function, let $a\in A,$ and let $L\in B.$ We say that $\ds\lim_{x\to a}f(x)=L$ if for all $\eps>0,$ there is a $\delta>0$ such that for all $x\in A,$ if $0<\rho(x,a)<\delta,$ then $\tau(f(x),L)<\eps.$

But we can rephrase these definitions in terms of open balls to be more geometrically intuitive. This works more effectively for the definition of continuity instead of the limit of a function:

Definition: Let $(A,\rho)$ and $(B,\tau)$ be metric spaces, let $f: A\to B$ be a function, and let $a\in A.$ We say that $f$ is continuous at $a$ if for all $\eps>0,$ there is a $\delta>0$ such that $f(B_\delta^\rho(a))\subseteq B_\eps^\tau(f(a)).$

We then get a very nice characterization of continuity in terms of limits of sequences:

Theorem: Let $(A,\rho)$ and $(B,\tau)$ be metric spaces, let $f: A\to B$ be a function, and let $a\in A.$ Then $f$ is continuous at $a$ if and only if for every sequence $\{x_n\}_{n=1}^\infty$ in $A,$ if $\ds\lim_{n\to \infty}x_n = a,$ then $\ds\lim_{n\to \infty}f(x_n) = f(a).$

We define $f$ to be continuous if it is continuous at every point of its domain, $A.$ We then see that $f$ is continuous if and only if for every convergent sequence $\{x_n\}_{n=1}^\infty,$ $ \ds\lim_{n\to\infty}f(x_n) = f\big(\lim_{n\to\infty} x_n\big).$

We get another, somewhat surprising, characterization of continuity purely in terms of open sets. This shows that continuity is actually a topological rather than a metric property.

Theorem: Let $(A,\rho)$ and $(B,\tau)$ be metric spaces, and let $f: A\to B$ be a function. Then $f$ is continuous if and only if for every open set $\mathcal O\subseteq B,$ the inverse image set $f^{-1}(\mathcal O)$ is open in $A.$

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