For open subsets $A\subseteq\mathbb{R}^n$, no choice is needed. Indeed, note that if $B\subset\mathbb{R}^n$ is any closed ball, there is a continuous surjection from an interval to $B$. Concatenating infinitely many such space-filling curves onto smaller and smaller balls around a point $a$, we get a single continuous path $\gamma:[0,1]\to A$ such that a map $f$ on $A$ is continuous at $a$ iff $f\circ\gamma$ is continuous at $0$.
Let us now discuss more general spaces; the upshot is that the axiom of choice is required to prove these results in general. I will say a space $A$ is pseudopath-generated if it satisfies your condition: that is, a map $f:A\to Y$ is continuous at a point $a\in A$ iff for any map $\gamma:[0,1]\to A$ such that $\gamma(0)=a$ and $\gamma$ is continuous at $0$, $f\circ\gamma$ is continuous at $0$. I will say a space $A$ is path-generated if the same condition holds but with $\gamma$ required to be continuous on all of $[0,1]$.
Note that the result you linked regarding first countable locally path-connected spaces actually says that they are path-generated, not just pseudopath-generated. Indeed, pseudopath-generation doesn't really have anything to do with paths, since continuity of $\gamma$ at a single point is a very weak condition. A similar argument shows that in fact every first countable space is pseudopath-generated.
The statement that every first countable locally path-connected space is path-generated is equivalent to countable choice. Let me first observe that countable choice is equivalent to the following (seemingly weaker) statement.
$(*)$ Given a countable family of $(X_n)_{n\in\mathbb{N}}$ of nonempty sets, there exists an infinite subset $S\subseteq\mathbb{N}$ and a choice function for $(X_n)_{n\in S}$.
To prove countable choice from $(*)$, let $(Y_n)_{n\in\mathbb{N}}$ be a countable family of nonempty sets and let $X_n$ be the set of choice functions for the initial segment $(Y_m)_{m<n}$. Then $(*)$ gives a choice function $f$ on an subsequence of $(X_n)$, which then gives a choice function $g$ for all of $(Y_n)$ (let $g(m)=f(n)(m)$ where $n$ is minimal in the domain of $f$ such that $f(n)(m)$ is defined).
Now here is a sketch of a proof that path-generation of first countable locally path-connected spaces implies $(*)$. Let $(X_n)$ be a countable family of nonempty sets. Let $*$ be some point that is not an element of any $X_n$ and let $Y_n=X_n\cup\{*\}$. Let $G$ be the graph with vertex set $\mathbb{N}$ and an edge from $n$ to $n+1$ for each element of $Y_n$. Consider this graph as a topological space, not with the weak topology but with the natural metric that makes each edge a path of length 1. Let $A=G\cup\{\infty\}$, where a neighborhood of $\infty$ must contain all sufficiently large vertices and edges between them in $G$.
Then $A$ is first countable and locally path-connected (for local path-connectedness at $\infty$, we use the fact that we can get a path from any vertex of $G$ to $\infty$ by infinitely concatenating the paths from $n$ to $n+1$ indexed by $*$). Now consider the map $f:A\to[0,1]$ defined as follows: $f(x)=0$ if $x$ is a vertex of $G$ or $\infty$, or is on one of the paths labelled by $*$. On each path labelled by an element of $X_n$, $f$ starts at $0$, goes up to $1$, and then goes back down to $0$.
This map $f$ is not continuous at $\infty$, since every neighborhood of $\infty$ contains paths labelled by elements of $X_n$. But to witness this discontinuity with a path in $A$, we would need a path $\gamma:[0,1]\to A$ which passes through paths corresponding to elements of $X_n$ for arbitrarily large $n$. Such a path would give a choice function on a subsequence of $(X_n)$ (for instance, for each $n$ such that $\gamma$ goes into the path corresponding to an element of $X_n$, choose the least rational point in $[0,1]$ with respect to some well-ordering of the rationals which maps into the path corresponding to an element of $X_n$, and use that to choose an element of $X_n$).
Let me now return to pseudopath-generated spaces. Everything in the argument above still works if $A$ is known only to be pseudopath-generated, except for the very last step where we use the path $\gamma$ to get a choice function on a subsequence of $(X_n)$. If $\gamma$ is not assumed to be continuous on $[0,1]$, then we do not have a canonical way to choose one particular element of $X_n$ which it hits. However, if we assume that $\mathbb{R}$ can be well-ordered, then the argument still works and proves countable choice. In particular, since ZF+"there exists a well-ordering of $\mathbb{R}$" cannot prove countable choice, this implies that ZF cannot prove every first countable locally path-connected space is pseudopath-generated.
By similar arguments we can prove the following are equivalent over ZF:
- All first-countable spaces are pseudopath-generated.
- If $(X_n)$ is any family of nonempty sets then there exists a function $f$ on $\mathbb{N}\times\mathbb{R}$ such that for each $n$, there exists $r\in\mathbb{R}$ such that $f(n,r)\in X_n$.
Statement 2 follows from countable choice and cannot be proved in ZF, but I suspect it is strictly weaker than countable choice.
