Building continuous functions

You do not need to test every function point by point. Continuous functions combine into continuous functions.

If f and g are continuous at c, so are:

• f + g, f – g and f × g;
• f / g, provided g(c) ≠ 0 (you cannot divide by zero);
• k f for any constant k.

From these rules:

• Every polynomial is continuous everywhere: it is built from x and constants by adding and multiplying.
• A rational function p(x)/q(x) is continuous everywhere except where q(x) = 0, where it is not even defined.

Trig functions: sin x and cos x are continuous everywhere. tan x = sin x / cos x, so it breaks only where cos x = 0, at the odd multiples of π/2. cosec x and cot x break where sin x = 0, at the multiples of π.

Composition: if g is continuous and f is continuous, then f(g(x)) is continuous. So cos(x² + 1), e^(sin x) and |x – 3| are all continuous everywhere.

So the usual way to answer 'Is this function continuous?' is to name its building blocks and quote the rule, for example: '|x| and |x – 1| are continuous everywhere, so their sum is continuous everywhere.'