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My professor introduced the following definition:

Let $f_n:(a,b) \rightarrow \mathbb{R}, n \in \mathbb{N}$ be a sequence of differentiable functions such that:

  1. $\sum_{n=1}^\infty f_n(c)$ converges for some $c \in (a,b)$
  2. $g := \sum_{n=1}^\infty f_n^{'}$ is uniformly convergent on $(a,b)$

Then $f:= \sum_{n=1}^\infty f_n$ is uniformly convergent on $(a,b)$ and $f^{'}=g$ on $(a,b)$.

What does it mean for a function to be uniformly convergent on an interval. I thought uniform convergence only applied to sequences of functions? How can a singular function be uniformly convergent? Even defined as a sum, the sum isn't a sequence but rather a limit of partial sums. How does it make sense for a limit to converge?

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  • $\begingroup$ $g$ is uniformly convergence means the partial sums are uniformly convergent (to $g$). $\endgroup$ Commented Apr 26, 2024 at 18:27
  • $\begingroup$ It is not the function $f$ that is uniformly convergent. The series $\sum_{n=1}^\infty f_n$ is uniformly convergent. It is perhaps confusing (but nevertheless common) to use the same letter for the series and for the sum of the series. $\endgroup$ Commented Apr 26, 2024 at 18:36
  • $\begingroup$ Uniform convergence on an interval means that this sequence is uniformly convergent on (a, b) because it can be non-uniformly convergent or even divergent outside of this interval. Because the domain is an (a, b) interval think of it as normal uniform convergence. Just remember to put $x∈(a,b)$ everywhere needed in your proof. $\endgroup$ Commented Apr 26, 2024 at 20:41

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The more precise statement is that the sequence of partial sums $$F_n = \sum_{k=1}^n f_k$$ is uniformly convergent on $(a,b)$ as $n \to \infty$. To be uniformly convergent on $(a,b)$ means that:

$$\lim_{n\to \infty}\|f - F_n \|_{(a,b)} = \lim_{n\to\infty}\sup_{x \in (a,b)} |f(x) - F_n(x)| =0$$

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What does it mean for a function to be uniformly convergent on an interval?

Let $S$ be any non-empty set, let $(f_n: S \to \mathbb{C})_{n \in \mathbb{N}}$ be a sequence of functions and let $f: S \to \mathbb{C}$ be a function. We say that $f_n$ converges to $f$ uniformly on $S$ iff: $$\forall \epsilon > 0: \exists \delta > 0: \forall x \in S: |f_n(x)-f(x)| < \epsilon.$$ Now, you can just specialize to the case of an interval on the real line. If you have trouble understanding this definition, then it's necessary for you to revisit your textbook where this is discussed (most likely in some detail).

I though uniform convergence only applied to sequences of functions?

Indeed.

How can a singular function by uniformly convergent? Even defined as a sum, the sum isn't a sequence but rather a limit of partial sums. How does it make sense for a limit to converge?

This confusion just stems from the way the statement by your professor has been phrased. So, they said "Then, $f := \sum_{n=1}^{\infty} f_n$ is uniformly convergent". What they mean is that the sequence of partial sums converges uniformly to $f$ and we define: $$\left(\sum_{n=1}^{\infty} f_n\right)(x) := \lim_{n \to \infty} \sum_{n=1}^{\infty} f_n(x),$$ for each $x$ in the relevant domain. They're just being slightly sloppy with their phrasing.

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