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Let $X$, $A$, and $B$ be topological spaces. I'm interested in functions $f:A\rightarrow B$ that preserve continuous functions with domain $X$; that is functions $f$ such that

$f\circ a:X\rightarrow B$ is continuous whenever $a:X\rightarrow A$ is continuous.

Call such a function $X$-continuous. So $X$-continuity is vacuous if $X$ is discrete, and $X$ continuity coincides with continuity when $X=A$.

Can anybody point me to a reference where such functions are studied? (In particular, for fixed $A,B$, I want to know for which $X$ are continuity and $X$-continuity equivalent.)

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do you mean $a \circ f$ or $f \circ a$ ? as it is there is a mistake – Glougloubarbaki Jan 13 '14 at 1:14
Unless I'm missing something, for every continuous $f$ you'll have that $f\circ a$ is continuous as it's a composition of continuous functions. Are you looking for non-continuous $f$ where $f \circ a$ is still continuous? – fgp Jan 13 '14 at 1:19

I'm not sure what can be said in general, but here are a few basic observations.

  1. If $A$ is a quotient of $X$, i.e. there exists some quotient map $q : X \to A$, then $X$-continuity is equivalent continuity for maps $A \to B$. So, for instance, since every compact metric space is the continuous image of the Cantor set, Cantor set-continuity equals continuity in the category of compact metric spaces.

  2. If every point $a \in A$ has a neighbourhood homeomorphic to $X$, then $X$-continuity is equivalent to continuity for maps $A \to B$. So, for instance, $\mathbb{R}^n$-continuity equals continuity in the category of $n$-dimensional manifolds.

  3. If $X = [0,1]$, then a simple sufficient condition for a map $f : A \to B$ to be $X$-continuous is that the restriction of $f$ to every path-component of $A$ be continuous. This should help you construct discontinuous examples in cases where the path-components aren't open. For example, consider $X = [0,1]$ and $A =$ the toplogists sine curve; or $A = \mathbb{Q}$.

  4. Continuing in the spirit the last example, if $X$ is connected and $A$ is totally disconnected, then $X$-continuity is vacuous (since all maps $X \to A$ are constant) i.e. all maps $A \to B$ are $X$-continuous.

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