Question

What is a continuous function?

Answer 1

There are several definitions of continuous function, so I give you several...

Explanation:

Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).

Much more formally:

If `A sube RR` then `f(x):A->RR` is continuous iff

`AA x in A, delta in RR, delta > 0, EE epsilon in RR, epsilon > 0 :`

`AA x_1 in (x - epsilon, x + epsilon) nn A, f(x_1) in (f(x) - delta, f(x) + delta)`

That's rather a mouthful, but basically means that `f(x)` does not suddenly jump in value.

Here's another definition:

If `A` and `B` are any sets with a definition of open subsets, then `f:A->B` is continuous iff the pre-image of any open subset of `B` is an open subset of `A`.

That is if `B_1 sube B` is an open subset of `B` and `A_1 = { a in A : f(a) in B_1 }`, then `A_1` is an open subset of `A`.