Continuity Invariants

Using all of the previous limit laws that we have proved, we can show that continuity is preserved by the standard arithmetic operations. Let $c, r, a, b \in \mathbb{R}$ be any constants with $b > 1$. If $f(x)$ and $g(x)$ are continuous, then the following are also continuous.

• $(f + g)(x)$
• $(f - g)(x)$
• $(c \cdot f)(x)$
• $(f \cdot g)(x)$
• $(1 / f)(x)$
• $(f / g)(x)$
• $(f^r)(x)$
• $(a^f)(x)$
• $(f^g)(x)$
• $(\log_b f)(x)$
• $(f \circ g)(x)$

In particular, since $f(x) = x$ is continuous, this means that all polynomials are continuous. Also, functions of the form $\displaystyle \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials are continuous. This is an important fact used often in calculus.

This provides insight as to why continuity is so important. Continuous functions behave in an intuitive way. A well-behaved function typically refers to a smooth, continuous function.