I'm working through a book called "Introduction to Topology" and I'm currently working on a chapter regarding metric spaces and continuity. This is how my book defines continuity at a point:
Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $a\in X$. A function $f: X\to Y$ is said to be continuous at the point $a\in X$ if given $\epsilon \gt 0$, there is a $\delta \gt 0$, such that $$d'(f(x),f(a)) \lt \epsilon$$ whenever $x\in X$ and $$d(x,a)\lt \delta$$
My question is this: is it possible that a function may be continuous in the metric spaces $(X,d)$ and $(Y, d')$ but not be continuous if one of the distance functions $d$ or $d'$ is changed to a different distance function?
In other words, does the continuity of a function depend on the distance functions used to "measure" it?