Something connected with Arzelà-Ascoli theorem

Let $X$ be a Polish space. Assume that $(C_m)_{m\in\mathbb{N}}$ is an increasing sequence of compact subsets of $X$ and denote $C=\bigcup_{m}C_m$. Let $\{f_n:n\in\mathbb{N}\}$ be a family of functions from $X$ in $\left[0,1\right]$, which is equicontinuous on compact subsets of $X$.

By the Arzelà-Ascoli theorem we can find a subsequence $(f_{k_n})_{n\in\mathbb{N}}$ convergent to a function $f$ uniformly on the set $C_m$, for any $m \in \mathbb{N}$.

Naturally $f$ is continuous on each set $C_m$, but a function with this property need not to be continuous on the set $C$.

Can one choose a subsequence in such a way that the limit be a continuous function on $C$?

As for me, this concept is too optimistic, but it was used in the paper Remarks on Ergodic Conditions for Markov Processes on Polish Spaces by Stettner (page 110, step 3).

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Your function $f$ is in fact continuous on $C$. The reason is that a function defined on a metric space is continuous iff its restriction to any compact set is continuous. Here are some more details.
Let $(x_i)$ be a sequence in $C$ converging to $x\in C$. Then $K=\{ x\}\cup \{ x_i;\; i\in\mathbb N\}$ is a compact subset of $X$. So $(f_n)$ is equicontinuous on $K$, and since $(f_{k_n})$ converges pointwise to $f$ on $K$ (because $K\subset C$), this implies that $f(x_i)\to f(x)$.