real analysis - Prove that the graph of a continuous function on $(-infty, infty)$ is completely determined once one knows a certain countable set of points on it.

A question from Introduction to Analysis by Arthur Mattuck:

Prove that the graph of a continuous function on $(-\infty, \infty)$ is completely determined once one knows a certain countable set of points on it.

I have no idea.

Answer

Hint: $\mathbb Q$ is countable and dense in $\mathbb R$.