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A map $\varphi:X\to Y$ between topological spaces is called continuous on paths if for every continuous path $\gamma :[0,1]\to X$ the function $\varphi \circ \gamma :[0,1]\to Y$ is also continuous. Every continuous map is also continuous on paths.

I am looking to prove the following claim:

If $X$ is locally path-connected, then a map $\varphi:X\to Y$ which is continuous on paths is also continuous.

I have found a proof of this which also assumes that $X$ is first countable (See: Continuity on paths implies continuity on space?). Is there a counter-example when $X$ is not first countable?

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Here is a counterexample. Let $D$ be an uncountable set with the discrete topology and let $W=D\times [0,1]/D\times\{0\}$ be the cone on $D$; we will write $0\in W$ for the cone point. Let $X$ have the same underlying set as $W$ but have the following coarser topology: an open set $U\subseteq W$ is open in $X$ iff either $0\not\in U$ or $U$ contains all of $\{d\}\times[0,1]$ for all but countably many $d\in D$.

It is easy to see that $X$ is locally path-connected. However, I claim that continuity on paths does not determine continuity on $X$. It suffices to give a set $A\subset X$ whose preimage under any path is closed but which is not closed in $A$ (since then you can consider $\varphi:X\to \{0,1\}$ which is the characteristic function of $A$, with $\{0,1\}$ topologized such that $\{1\}$ is closed and $\{0\}$ is not). An example of such a set is $A=D\times\{1\}$. Indeed, $A$ is not closed since $0\in\overline{A}$. However, any path $f:[0,1]\to X$ intersects $\{d\}\times(0,1]$ for only countably many values of $d$ (for instance, because the image of $f$ is contained in the closure of the countable set $f([0,1]\cap\mathbb{Q})$). Since $S\times\{1\}$ is closed in $X$ for any countable subset $S\subset D$, it follows that $f^{-1}(A)$ is closed.

For more discussion (including a counterexample that is additionally a sequential space), see my answer at Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$?.

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