Question

What are some examples of continuous functions?

Answer 1

(1) `f(x) = x^2`,
(2) `g(x) = sin(x)`
(3) `h(x) = 3x+1`

Explanation:

A function is continuous, intuitively, if it can be drawn (i.e. graphed) without having to lift the pencil (or pen) from the paper. That is, approaching any point x, in the domain of the function from the left, i.e. x-`epsilon`, as `epsilon ->` 0, yields the same value as approaching the same point from the right, i.e. x+`epsilon`, as ε→ 0. This is the case with each of the functions listed.

It would not be the case for the function d(x) defined by: `d(x) = 1`, if x `>=` 0, and `d(x) = -1`, if x < 0. That is, there is a discontinuity at 0, as approaching 0 from the left, one has the value -1, but approaching from the right, one has the value 1.