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

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?

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How do you define discrete? – Stefan Hamcke Jul 28 '13 at 16:45
What's your definition of a discrete function? – tomasz Jul 28 '13 at 16:45
A function is discret if it's domain is at most countable – Daniela Diaz Jul 28 '13 at 16:45
In my first year of study we learned that intuitively a function is continuous if doesn't make jumps. Consider the domain $\Bbb Q\cap([-1,0)\cup(0,1])$ and the function $f(x)=\begin{cases} 0, \text{ if } x<0\\ 1, \text{ if } x>0 \end{cases}\ .\ \$ The domain is countable, but the hole at $0$ is what enables the function to be continuous. If we added the irrational numbers to the domain to eliminate the holes, then there would be no way to define $f(0)$ so that $f$ be continuous. So having holes and being continuous is not a contradiction. Sometimes you need holes in order to have continuity. – Stefan Hamcke Jul 28 '13 at 16:59

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.