Algebra of Continuous Functions

Continuity is preserved under the ordinary arithmetic operations. This is what lets complicated functions be judged continuous by breaking them into simple, already-known continuous pieces.

The algebra rules. If $f$ and $g$ are both continuous at a point $x = a$, then so are:

1. their sum $f + g$ and difference $f - g$;
2. any scalar multiple $c\,f$ (for a constant $c$), and their product $f \cdot g$;
3. their quotient $\dfrac{f}{g}$ — provided $g(a) \neq 0$ (a quotient can only fail continuity where the denominator vanishes).

Composite functions. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite $f \circ g$, given by $(f\circ g)(x) = f\big(g(x)\big)$, is continuous at $a$. For example $|x^2 - 4|$ is continuous everywhere, being the modulus function (continuous everywhere) applied to the polynomial $x^2 - 4$ (continuous everywhere).

Method — deciding continuity of a built-up function:

1. Break the function into its building blocks (polynomials, modulus, rational parts, exponentials, logs).
2. Note where each block is continuous, using the standard results of §4.
3. Apply the algebra rules above: the function is continuous wherever all its blocks are continuous and no denominator is zero.
4. The only points needing a closer, from-scratch check are where a denominator vanishes or where a piecewise rule changes.

Illustration. Consider $f(x) = x^2 + |x - 1|$. The term $x^2$ is a polynomial (continuous everywhere) and $|x-1|$ is a modulus of a polynomial (continuous everywhere, by the composite rule). By the sum rule, $f$ is continuous at every real number — no point-by-point limit test is needed anywhere.