# Existence of a continuous function from $R$ to a countable collection of points of $R$.

Let $(x_i)_i$ be a sequence of real numbers, where finite numbers of $x_i$'s must be unequal. Let us construct a function $f:R→X$ where the set $X$ denote the elements of the aforesaid sequence.

Is there any function $f$ like above is continuous, that is is there any function $f$ from real to any countable subset of $R$? I know that if the sequence is constant sequence then it is trivial, but what about countable points of $R$?

• $\mathbb R$ is connected. Any continuous function must map connected sets to connected sets. If the image of $f$ contains distinct points $x$ and $y$, it must contain the entire interval between $x$ and $y$. – Bungo Nov 7 '17 at 22:51
• Yah this trivial. And thanks for your answer. @Bungo – abcdmath Nov 7 '17 at 22:57