Contrapositive of uniform limit theorem for series

Question

I am looking for the contrapositive of the uniform limit theorem, especially in regards to series. For functions we have (and I am quoting from another post on this site)

If a sequence of continuous functions $\left(f_n(x)\right)_{n=1}^\infty$ converges to a function $f(x)$ uniformly [on $D$], then $f(x)$ is continuous [on $D$].

This quote doesn't state any domain of all $f_n(x)$ and $f(x)$ and hence I added the domain $D$. My guess for a contrapositive would be

1. If $f(x)$ is discontinuous at a point in $D$, then $\left(f_n(x)\right)_{n=1}^\infty$ does not converge uniformly to $f(x)$ on $D$.

I am unsure if we take the contrapositive of the fact that all $f_n(x)$ are continuous. Anyway, what I want to ask about is the following, similar statement (inspired by this post):

Let $\left(f_n(x)\right)_{n=1}^\infty$ be a sequence of continuous functions on $D$ and define $$s_n(x)=\sum_{k=1}^{n} f_k(x).$$ If the sequence $\left(s_n(x)\right)_{n=1}^\infty$ converges uniformly to $s(x)=\sum_{n=1}^{\infty} f_n(x)$ on $D$, then $s(x)$ is continuous on $D$.

Now, here I am confused about the contrapositive, because other conditions may affect the conclusion of the implication. Here's what I mean. My guess at a contrapositive would be:

2. If $s(x)$ is discontinuous at a point in $D$, then $\left(s_n(x)\right)_{n=1}^\infty$ does not converge uniformly on $D$.

However, suppose $s(x)$ does not converge for some $x\in D$ or that some of the $f_n(x)$ are discontinuous. Then we don't have uniform convergence either, or? Should we include that in the contrapositive in 2?

Answer 1

Let $C(A)$ be the set of all real-valued functions that are continuous on $A\subseteq\mathbb{R}$. Denote by $\mathbb{R}^A$ and $C(A)^\mathbb{N}$ the set of all functions $h:A\to\mathbb{R}$ and the set of all sequences in $C(A)$ respectively (the latter is all functions $g:\mathbb{N}\to C(A)$ with $n \mapsto f_n$, i.e. $(g(1),g(2),\ldots)=(f_1,f_2,\ldots)\in C(A)^\mathbb{N}$). Then

Theorem. If a sequence $(f_n)$ of continuous functions $f_n: A\to\mathbb{R}$ converges uniformly on $A\subseteq\mathbb{R}$ to $f:A\to\mathbb{R}$, then $f$ is continuous on $A$.

reads

$$(\forall A\subseteq\mathbb{R})(\forall g\in C(A)^\mathbb{N})(\forall h\in \mathbb{R}^A)(g \text{ converges uniformly to } h\text{ on } A \implies h\in C(A))$$

The contrapositive of the implication is now trivial and applies to series as well.