What is the difference between a discrete function and a continuous function

Question

Intuitively it seems that both concepts should be disjoint because if a function is discrete then it has some holes on it and if a function is continuous then it doesn't have holes. But now I'm not sure because, from my understanding, a function may be continuous at $x_{0}$ if $x_{0}$ is an accumulation point in its domain such that $\lim_{x\to x_0}f=f(x_{0})$. So for example the function $f:\mathbb{Q}\to \mathbb{R}$ such that $f(x)=x$ is such that $\lim _{x\to x_0}f=x_{0}=f(x_{0})$ and then $f$ is continuous at any point in its domain but also it's discrete. What I'm a missing?

Answer 1

I hope you mean "discrete function" to be a function with discrete (i.e. at most countable) domain...

For infinite, you may use the equivalent definition of continuity by Heine: "$A$ is limit of $f$ in accumulation point $a$ iff for each sequence $a_n$ tending to $a$, the limit $f(a_n)$ is $A$", so actually a discrete function can be continuous.