continuous function between discrete topological space and real numbers set Let $X$ be a topological space, with the topology being the discrete one. I have to find the continuous functions between $X$ and $\mathbb{R}$ with the usual topology. As an idea, I thought that I could use the fact that a discrete space is disconnected and a continuous functions takes connceted sets into connected sets. So, the only connected subsets of $X$ are the singletons. Is that a helpful idea?Thank you very much!($X$ is a nonempty set).
 A: Recall that a function $f:X\to Y$ is continuous if and only if for all $U\subseteq Y$ open, $f^{-1}[U]$ is open. 
In your case, $X$ is a discrete space and $Y=\mathbb{R}$. Take a function $g:X\to Y$ and $U\subseteq \mathbb{R}$ an open set. What we can say about $g^{-1}[U]$? In the discrete space, every set is open. Thus, $g^{-1}[U]\subseteq X$ is open and therefore, $g$ is continuous. Note that $g$ was an arbitrary function!
A: Your idea is not a bad one, but it is perhaps "too sophisticated" to find the answer. Try instead to go back to the definition of the discrete topology and the definition of a continuous function. What does a function $f \colon X \to \mathbb{R}$ have to satisfy in order to be continuous, when you know that $X$ is discrete? 
A: If $X$ is discrete reals and $Y$ is standard reals, then 


*

*all functions from $X$ to $Y$ are continuous  the definition of continuity gives this immediately).

*the only continuous functions from $Y$ to $X$ are the constant ones (by the connectedness argument: the only connected subspaces of $X$ are the singletons).

