Let us get back to swapping limits and expand on Chapter 6. Let \(\{ f_n \}_{n=1}^\infty\) be a sequence of functions \(f_n \colon X \to Y\) for a set \(X\) and a metric space \(Y\text{.}\) Let \(f \colon X \to Y\) be a function and for every \(x \in X\text{,}\) suppose
The question is: If \(f_n\) are all continuous, is \(f\) continuous? Differentiable? Integrable? What are the derivatives or integrals of \(f\text{?}\) For example, for continuity of the pointwise limit of a sequence of functions \(\{ f_n \}_{n=1}^\infty\text{,}\) we are asking if
So pointwise convergence is not enough to preserve continuity (nor even boundedness). For that, we need uniform convergence. Let \(f_n \colon X \to Y\) be functions. Then \(\{f_n\}_{n=1}^\infty\)converges uniformly to \(f\) if for every \(\epsilon > 0\text{,}\) there exists an \(M\) such that for all \(n \geq M\) and all \(x \in X\text{,}\) we have
A series \(\sum_{n=1}^\infty f_n\) of complex-valued functions converges uniformly if the sequence of partial sums converges uniformly, that is, if for every \(\epsilon > 0\text{,}\) there exists an \(M\) such that for all \(n \geq M\) and all \(x \in X\text{,}\)
The simplest property preserved by uniform convergence is boundedness. We leave the proof of the following proposition as an exercise. It is almost identical to the proof for real-valued functions.
Proposition11.2.2.
Let \(X\) be a set and \((Y,d)\) a metric space. If \(f_n \colon X \to Y\) are bounded functions and converge uniformly to \(f
\colon X \to Y\text{,}\) then \(f\) is bounded.
If \(X\) is a set and \((Y,d)\) is a metric space, then a sequence \(f_n \colon X
\to Y\) is uniformly Cauchy if for every \(\epsilon > 0\text{,}\) there is an \(M\) such that for all \(n, m \geq M\) and all \(x \in X\text{,}\) we have
The notion is the same as for real-valued functions. The proof of the following proposition is again essentially the same as in that setting and is left as an exercise.
Proposition11.2.3.
Let \(X\) be a set, \((Y,d)\) be a metric space, and \(f_n \colon X \to Y\) be functions. If \(\{ f_n \}_{n=1}^\infty\) converges uniformly, then \(\{f_n\}_{n=1}^\infty\) is uniformly Cauchy. Conversely, if \(\{f_n\}_{n=1}^\infty\) is uniformly Cauchy and \((Y,d)\) is Cauchy-complete, then \(\{f_n\}_{n=1}^\infty\) converges uniformly.
We call \(\snorm{\cdot}_X\) the supremum norm or uniform norm, and the subscript denotes the set over which the supremum is taken. Then a sequence of functions \(f_n \colon X \to \C\) converges uniformly to \(f \colon X \to \C\) if and only if
For a compact metric space \(X\text{,}\) the uniform norm is a norm on the vector space \(C(X,\C)\text{.}\) We leave it as an exercise. While we will not need it, \(C(X,\C)\) is in fact a complex vector space, that is, in the definition of a vector space we can replace \(\R\) with \(\C\text{.}\) Convergence in the metric space \(C(X,\C)\) is uniform convergence.
We will study a couple of types of series of functions, and a useful test for uniform convergence of a series is the so-called Weierstrass \(M\)-test.
Theorem11.2.4.Weierstrass \(M\)-test.
Let \(X\) be a set. Suppose \(f_n \colon X \to \C\) are functions and \(M_n > 0\) numbers such that
\begin{equation*}
\sabs{f_n(x)}\leq M_n \quad \text{for all } x \in X,
\qquad \text{and} \qquad
\sum_{n=1}^\infty M_n
\quad \text{converges}.
\end{equation*}
Another way to state the theorem is to say that if \(\sum_{n=1}^\infty \snorm{f_n}_X\) converges, then \(\sum_{n=1}^\infty f_n\) converges uniformly. Note that the converse of this theorem is not true. Applying the theorem to \(\sum_{n=1}^\infty \sabs{f_n(x)}\text{,}\) we see that this series also converges uniformly. So the series converges both absolutely and uniformly.
Proof.
Suppose \(\sum_{n=1}^\infty M_n\) converges. Given \(\epsilon > 0\text{,}\) we have that the partial sums of \(\sum_{n=1}^\infty M_n\) are Cauchy so there is an \(N\) such that for all \(m, n \geq N\) with \(m \geq n\text{,}\) we have
converges uniformly on \(\R\text{.}\) See Figure 11.2. This series is a Fourier series, and we will see more of these in a later section. Proof: The series converges uniformly because \(\sum_{n=1}^\infty \frac{1}{n^2}\) converges and
converges uniformly on every bounded interval. This series is a power series that we will study shortly. Proof: Take the interval \([-r,r] \subset \R\) (every bounded interval is contained in some \([-r,r]\)). The series \(\sum_{n=0}^\infty \frac{r^n}{n!}\) converges by the ratio test, so \(\sum_{n=0}^\infty \frac{x^n}{n!}\) converges uniformly on \([-r,r]\) as
Now we would love to say something about the limit. For example, is it continuous?
Proposition11.2.7.
Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, and suppose \((Y,d_Y)\) is Cauchy-complete. Suppose \(f_n \colon X \to Y\) converge uniformly to \(f \colon X \to Y\text{.}\) Let \(\{ x_k \}_{k=1}^\infty\) be a sequence in \(X\) and \(x \coloneqq \lim_{k \to \infty} x_k\text{.}\) Suppose
First we show that \(\{ a_n \}_{n=1}^\infty\) converges. As \(\{ f_n \}_{n=1}^\infty\) converges uniformly it is uniformly Cauchy. Let \(\epsilon > 0\) be given. There is an \(M\) such that for all \(m,n \geq M\text{,}\) we have
\begin{equation*}
d_Y\bigl(f_n(x_k),f_m(x_k)\bigr) < \epsilon \qquad \text{for all } k .
\end{equation*}
Note that \(d_Y(a_n,a_m) \leq
d_Y\bigl(a_n,f_n(x_k)\bigr) +
d_Y\bigl(f_n(x_k),f_m(x_k)\bigr) +
d_Y\bigl(f_m(x_k),a_m\bigr)\) and take the limit as \(k \to \infty\) to find
We obtain an immediate corollary about continuity. If \(f_n\) are all continuous then \(a_n = f_n(x)\) and so \(\{ a_n \}_{n=1}^\infty\) converges automatically to \(f(x)\) and so we do not require completeness of \(Y\text{.}\)
Corollary11.2.8.
Let \(X\) and \(Y\) be metric spaces. If \(f_n \colon X \to Y\) are continuous functions such that \(\{ f_n \}_{n=1}^\infty\) converges uniformly to \(f \colon X \to Y\text{,}\) then \(f\) is continuous.
The converse is not true. Just because the limit is continuous does not mean that the convergence is uniform. For example: \(f_n \colon (0,1) \to \R\) defined by \(f_n(x) \coloneqq x^n\) converge to the zero function, but not uniformly. However, if we add extra conditions on the sequence, we can obtain a partial converse such as Dini’s theorem, see Exercise 6.2.10.
In Exercise 11.2.3 the reader is asked to prove that for a compact \(X\text{,}\)\(C(X,\C)\) is a normed vector space with the uniform norm, and hence a metric space. We have just shown that \(C(X,\C)\) is Cauchy-complete: Proposition 11.2.3 says that a Cauchy sequence in \(C(X,\C)\) converges uniformly to some function, and Corollary 11.2.8 shows that the limit is continuous and hence in \(C(X,\C)\text{.}\)
Corollary11.2.9.
Let \((X,d)\) be a compact metric space. Then \(C(X,\C)\) is a Cauchy-complete metric space.
Suppose \(f_n \colon [a,b] \to \C\) are Riemann integrable and suppose that \(\{ f_n \}_{n=1}^\infty\) converges uniformly to \(f \colon [a,b] \to \C\text{.}\) Then \(f\) is Riemann integrable and
\begin{equation*}
\int_a^b f = \lim_{n\to \infty} \int_a^b f_n .
\end{equation*}
Since the integral of a complex-valued function is just the integral of the real and imaginary parts separately, the proof follows directly by the results of Chapter 6. We leave the details as an exercise.
Corollary11.2.12.
Suppose \(f_n \colon [a,b] \to \C\) are Riemann integrable and suppose that
The swapping of integral and sum is possible because of uniform convergence, which we have proved before using the Weierstrass \(M\)-test (Theorem 11.2.4).
We remark that we can swap integrals and limits under far less stringent hypotheses, but for that we would need a stronger integral than the Riemann integral. E.g. the Lebesgue integral.
Subsection11.2.3Differentiation
Recall that a complex-valued function \(f \colon [a,b] \to \C\text{,}\) where \(f(x) = u(x)+i\,v(x)\text{,}\) is differentiable, if \(u\) and \(v\) are differentiable and the derivative is
The proof of the following theorem is to apply the corresponding theorem for real functions to \(u\) and \(v\text{,}\) and is left as an exercise.
Theorem11.2.14.
Let \(I \subset \R\) be a bounded interval and let \(f_n \colon I \to \C\) be continuously differentiable functions. Suppose \(\{ f_n' \}_{n=1}^\infty\) converges uniformly to \(g \colon I \to \C\text{,}\) and suppose \(\{ f_n(c) \}_{n=1}^\infty\) is a convergent sequence for some \(c \in I\text{.}\) Then \(\{ f_n \}_{n=1}^\infty\) converges uniformly to a continuously differentiable function \(f \colon I \to \C\text{,}\) and \(f' = g\text{.}\)
Uniform limits of the functions themselves are not enough, and can make matters even worse. In Section 11.7 we will prove that continuous functions are uniform limits of polynomials, yet as the following example demonstrates, a continuous function need not be differentiable anywhere.
Example11.2.15.
There exist continuous nowhere differentiable functions. Such functions are often called Weierstrass functions, although this particular one, essentially due to Takagi 1
Teiji Takagi (1875–1960) was a Japanese mathematician.
, is a different example than what Weierstrass gave. Define
Extend \(\varphi\) to all of \(\R\) by making it 2-periodic: Decree that \(\varphi(x) = \varphi(x+2)\text{.}\) The function \(\varphi \colon \R \to \R\) is continuous, in fact, \(\sabs{\varphi(x)-\varphi(y)} \leq \sabs{x-y}\) (why?). See Figure 11.3.
Figure11.3.The 2-periodic function \(\varphi\text{.}\)
As \(\sum_{n=0}^\infty {\left(\frac{3}{4}\right)}^n\) converges and \(\sabs{\varphi(x)} \leq
1\) for all \(x\text{,}\) by the \(M\)-test (Theorem 11.2.4),
If \(n > m\text{,}\) then \(4^n\delta_m\) is an even integer. As \(\varphi\) is 2-periodic we get that \(\gamma_n = 0\text{.}\)
As there is no integer between \(4^m(x+\delta_m) = 4^m x\pm\nicefrac{1}{2}\) and \(4^m x\text{,}\) then on this interval \(\varphi(t) = \pm t + \ell\) for some integer \(\ell\text{.}\) In particular, \(\abs{\varphi\bigl(4^m(x+ \delta_m)\bigr)-\varphi(4^mx)} =
\abs{4^mx\pm\nicefrac{1}{2}-4^mx} = \nicefrac{1}{2}\text{.}\) Therefore,
As \(m \to \infty\text{,}\) we have \(\delta_m \to 0\text{,}\) but \(\frac{3^m+1}{2}\) goes to infinity. So \(f\) cannot be differentiable at \(x\text{.}\)
Suppose \((X,d)\) is a compact metric space. Prove that the uniform norm \(\snorm{\cdot}_X\) is a norm on the vector space of continuous complex-valued functions \(C(X,\C)\text{.}\)
11.2.4.
Prove that \(f_n(x) \coloneqq 2^{-n} \sin(2^n x)\) converge uniformly to zero, but there exists a dense set \(D \subset \R\) such that \(\lim_{n\to\infty} f_n'(x) = 1\) for all \(x \in D\text{.}\)
Prove that \(\sum_{n=1}^\infty 2^{-n} \sin(2^n x)\) converges uniformly to a continuous function, and there exists a dense set \(D \subset \R\) where the derivatives of the partial sums do not converge.
11.2.5.
Prove that \(\snorm{f}_{C^1} \coloneqq \snorm{f}_{[a,b]}+\snorm{f'}_{[a,b]}\) is a norm on the vector space of continuously differentiable complex-valued functions \(C^1\bigl([a,b],\C\bigr)\text{.}\)
