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Give an example of a function which is continuous with respect to some metric but not continuous with respect to some other metric.

(I take the identity function from (R,d_1) to (R,d_2) where d_1 is discrete metric and d_2 is usual metric. It is continuous on R. I am trying to replace the metric d_1 only (not d_2) by some other metric, say d_3, so that the identity function from (R,d_3) to (R,d_2) becomes discontinuous.)

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    Consider the discrete metric: in such a metric all functions are continuous! – 2017-01-07

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Let $X=\mathbb R$; consider metrics $d_1,d_2$ with $$d_1(x,y)=|x-y|$$ and $$d_2(x,y)=\begin{cases}1 &\text{if}\ x\ne y \\ 0 &\text{if}\ x=y\end{cases}$$ If $\iota:X\to X$ is the identity function, then $$\iota:(X,d_2)\to(X,d_1)$$ is continuous at all points $y\in X$. Given any $\varepsilon>0$, take $\delta=1$; then for any $x\in X$, $d_2(x,y)<\delta$ $\implies$ $x=y$ $\implies$ $d_1(x,y)=0<\varepsilon$.

On the other hand $$\iota:(X,d_1)\to(X,d_2)$$ is not continuous at any point $y\in X$. Given $\varepsilon=1$ and any $\delta>0$, we can always find $x$ such that $0<|x-y|=d_1(x,y)<\delta$; then $d_2(x,y)=1\ge\varepsilon$.