Show that every mapping $f: X \to Y$ is continuous where $(X,d)$ is a metric space with the discrete metric Let $(X,d)$ be a metric space with the discrete metric and let $(Y,d')$ be an arbitrary metric space. Show that every mapping $f: X\to Y$ is continuous.

I got a bit confused about this, initially, I thought that it would come quite directly from the definition, but I'm not sure how to proceed since I'm not given $f$ to work with. Perhaps there is some other definition for continuity that I'm not familiar? I've used the one that states $\forall \varepsilon &gt;0 \ \exists \delta &gt;0$ s.t. $d(x,y) &lt; \delta \Longrightarrow |f(x)-f(y) | &lt; \varepsilon.$
Hint: What happens if you set $\delta=\frac12$?
Hint: What happens if you set $\delta=\frac12$?