Question

What is a continuous function?

Answer 1

A continuous function is a function that is continuous at every point in its domain.

That is `f:A->B` is continuous if `AA a in A, lim_(x->a) f(x) = f(a)`

Explanation:

We normally describe a continuous function as one whose graph can be drawn without any jumps. That's a good place to start, but is misleading.

An example of a well behaved continuous function would be `f(x) = x^3-x`

graph{x^3-x [-2.5, 2.5, -1.25, 1.25]}

In fact any polynomial is well defined everywhere and continuous.

A less obvious example of a continuous function is `f(x) = tan(x)`

graph{tan(x) [-10, 10, -5, 5]}

This appears to be discontinuous, with 'jumps' at `x = pi/2+n pi` but those values of `x` are excluded from the domain.

Similarly, the following function is continuous on its domain `(-oo, 0) uu (0, oo)`

`f(x) = x/abs(x)`

graph{x/abs(x) [-5, 5, -2.5, 2.5]}

If we add a definition of `f(0)` then this becomes a discontinuous function.

`f(x) = { (x/abs(x), "if x != 0"), (0, "if x = 0") :}`

graph{(y-x/abs(x))(x^2+y^2-0.002) = 0 [-5, 5, -2.5, 2.5]}

This `f(x)` fails the condition for continuity at the point `x=0`.

`lim_(x->0) f(x)` is not defined, let alone equal to `f(0)`