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In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge uniformly to some function over a set E, then we know that this function is continuous (or integrable) -- and then the limit of the integrals is equal to the integral of the limit function. (the stronger version of this integration theorem, the dominated convergence theorem, only requires pointwise convergence of the functions, in order to take the limit inside the integral.)

What else can we get from uniform convergence in the real variable case? Does it preserve differentiability? Or, in general it does not?

And, I think in the complex variable setting, uniform convergence preserves all of continuity, integrability, and differentiability.

...anything else to be aware of?

Thanks in advance,

User001
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2 Answers2

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Uniform convergence does not preserve differentiability in the real case. For example, the functions $f_n(x)=\sqrt{x^2+1/n}$ converge to $f(x)=|x|$ uniformly. Even worse, every continuous function on a closed bounded interval, including the nowhere-differentiable Weierstrass function, is the uniform limit of polynomials.

But in the complex case we have Cauchy integral formula that gives $f'$ as an integral involving $f$. This gives a bound on the derivative in terms of the function itself. Consequently, uniform convergence of holomorphic functions implies uniform convergence of their derivatives on every compact subset of the domain.

In both real and complex cases, uniform convergence of derivatives implies that the limit is differentiable. The difference is that in the complex case, the convergence of derivatives is automatic.

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You need stronger conditions than uniform convergence to ensure that

$$f_n(x) \to f(x) \implies f_n'(x) \to f'(x).$$

Here is a standard theorem found in virtually all real analysis books.

Suppose $(f_n)$ is a sequence of differentiable functions that converges pointwise at some point in $[a,b]$ and $(f_n')$ converges uniformly on $[a,b]$ to $g.$ Then $(f_n)$ converges uniformly to a differentiable function $f$ and $f_n'(x) \to g(x)$ for all $x \in [a,b].$

The following is an example where uniform convergence of $f_n \to f$ alone does not ensure $f_n' \to f'$.

Consider the sequence of functions $f_n:[a,b] \to \mathbf{R}$ with

$$f_n(x) = \frac{x}{1+nx^2}.$$

It's easy to show that $f_n$ has a maximum at $x = 1/\sqrt{n}$. Hence, on $[0,1]$,

$$f_n(x) \leqslant f_n\left(\frac1{\sqrt{n}}\right)= \frac1{2\sqrt{n}}.$$

Consequently $f_n(x) \rightarrow f(x) \equiv 0$ uniformly on $[0,1]$.

Also, $f_n$ is differentiable with

$$f'_n(x) = \frac{1-nx^2}{(1+nx^2)^2}.$$

However, $f'_n(0) \rightarrow 1 \neq f'(0).$

RRL
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