Function whose limit does not exist at all points There are functions which are discontinuous everywhere and there are functions which are not differentiable anywhere, but are there functions with domain $\mathbb{R}$ (or "most" of it) whose limit does not exist at every point? For example, $ f:\mathbb{R}\to\mathbb{N}, f(x) = $ {last digit of the decimal representation of $x$}. Is this even a valid function?

Now f(x) is define f:R→R by f(x)=x if x is rational and f(x)=-x if x is irrational. How to prove it limt (x approach to a) f(x) doesn't exist at a belongs to R with a not equal to 0.
Here limit doesn't go to constant value. How to prove it by using above theory.
