Section5.1Continuity of a Function of Several Variables
The concept of continuity of a map from a metric space to a metric space applies directly to a function of several variables---one needs to distinguish between continuity at a point, everywhere continuity, and uniform continuity. For two general metric space \(X\) and \(Y\text{,}\) there is often not much structure of the space \(\cC(X, Y)\) of maps from \(X\) to \(Y\) that are continuous everywhere (or continuous at a point) on \(X\text{;}\) for example, if \(\bff_{1},\bff_{2}\) are two such maps, there is usually no natural operation of \(\bff_{1}+\bff_{2}\) or \(\bff_{1}\cdot \bff_{2}\text{.}\) One natural operation for this general setting is composition: suppose \(\bff: X\mapsto Y\) and \(\bg:Y\mapsto Z\text{,}\) then \(\bg\circ\bff:X\mapsto Z\) is defined and when \(\bff\) is continuous at \(\bx_0\in X\) and \(\bff\) is continuous at \(\by_0:=\bff(\bx_0)\text{,}\) then \(\bg\circ\bff:X\mapsto Z\) is continuous at \(\bx_0\text{.}\)
However, when \(Y=\bbR^{m}\) for some \(m\text{,}\) then \(\bff_{1}+\bff_{2}\) makes a natural sense; in fact, for any scalar \(c_{1}, c_{2}\text{,}\)\(c_{1}\bff_{1}+c_{2}\bff_{2}\) is well defined and continuous on \(X\text{.}\) This makes \(\cC(X, \bbR^{m})\) a vector space.
Let \(D\) be a subset of \(\bbR^{n}\text{,}\)\(\bff :D\mapsto \bbR^{m}\) be a map defined on \(D\) and \(\bx_{0}\in D\text{.}\) Recall that \(\bff\) is continuous at \(\bx_{0}\text{,}\) if for any \(\epsilon \gt 0\text{,}\) there exists some \(\delta \gt 0\) such that for any \(\bx \in D\) with \(\Vert \bx -\bx_{0}\Vert \lt \delta\text{,}\) we have \(\Vert \bff (\bx) -\bff(\bx_{0})\Vert \lt \epsilon\text{.}\)
If we write out \(\bff(\bx)=(f_{1}(\bx),\cdots, f_{m}(\bx))\text{,}\) it’s clear that \(\bff\) is continuous at \(\bx_{0}\) iff each \(f_{i}(\bx)\text{,}\) for \(i=1,\cdots, m\text{,}\) is continuous at \(\bx_{0}\text{.}\)
The above property does not necessarily hold when the image space is infinitely dimensional. For example, if \(Y=l^2\) the space of sequences that are square summable. Suppose that \(\bff(\bx)=(f_{1}(\bx),\cdots, f_{m}(\bx),\cdots)\in l^2\) and each \(f_{i}(\bx)\text{,}\) for \(i=1,\cdots, \infty\text{,}\) is continuous at \(\bx_{0}\text{.}\) Does this imply that \(\bff(\bx)\) is continuous at \(\bx_{0}\text{?}\)
Let \(\eta:\bbR^+\mapsto \bbR^+\) be continuous such that \(\eta(t)=t\) for all \(0\le t\le 1\) and \(\eta(t)=0\) for all \(t\ge 2\text{.}\) Then \(f_m(t)=\frac{\eta(m t)}{\sqrt m}\) defines a continuous function on \(\bbR^{+}\) for each \(m\text{,}\) and \(\bff(t)=(\eta(t),\cdots, \frac{\eta(m t)}{\sqrt m},\cdots)\in l^2\) for each \(t\in \bbR^{+}\text{,}\) for, given any \(t\gt 0\text{,}\)\(\eta(mt)=0\) for all \(m\) such that \(mt\ge 2\text{,}\) so \(\bff(t)\) terminates after a finite number of terms. However, for \(t=1/N\text{,}\)
Example5.1.2.Some examples of continuous functions.
(a)
The functions defining the change of coordinate from polar coordinates to rectangular coordinates are continuous for \((r, \theta) \in \bbR^+ \times [0, 2\pi]\text{:}\)
\begin{align*}
x \amp = r \cos \theta\\
y \amp =r \sin \theta
\end{align*}
Part of the inverse, \(r=\sqrt{x^2+y^2}\text{,}\) is defined on \(\bbR^2\) and continuous there, but \(\theta\) as a function of \((x, y)\) is only defined on \(\bbR^2 \setminus\{\text{a ray from } 0\}\)--- one often uses the formula \(\theta=\tan^{-1}(\frac yx)\text{,}\) but it works only for \(x\gt 0\text{.}\)
Suppose \(\bff_1: U\mapsto \bbR^n\) and \(\bff_2: U\mapsto \bbR^n\) are two continuous functions from \(U\) to \(\bbR^n\text{,}\) then composition with the continuous inner product makes \(\bff_1(\bu)\cdot \bff_2(\bu)\) a continuous function from \(U\) to \(\bbR\text{.}\)
Let \((X, \Vert \cdot \Vert )\) be any normed vector space, then \(\bx \mapsto \Vert \bx\Vert \in \bbR\) is continuous from \(X\) to \(\bbR\text{.}\) This follows from using the triangle inequality to get
