Continuity of Standard Functions

Rather than apply the three-condition test from scratch every time, it is far quicker to know which standard functions are continuous and where. The following results are established once and then used freely.

• Polynomial functions — such as $f(x) = a_n x^n + \dots + a_1 x + a_0$ — are continuous at every real number. In particular constant functions $f(x)=c$ and the identity $f(x)=x$ are continuous everywhere.
• Rational functions $\dfrac{p(x)}{q(x)}$ (a ratio of two polynomials) are continuous at every point where the denominator is non-zero, i.e. at every point of their domain. They can only fail to be continuous where $q(x) = 0$.
• The modulus function $f(x) = |x|$ is continuous at every real number, including at $x = 0$, where its two pieces $-x$ (for $x<0$) and $x$ (for $x \ge 0$) meet at the common value $0$.
• The exponential function $f(x) = a^x$ (with $a > 0$), including $f(x) = e^x$, is continuous at every real number.
• The logarithmic function $f(x) = \log_a x$ (with $a > 0,\ a \neq 1$), including $f(x) = \ln x$, is continuous at every point of its domain, that is, for all $x > 0$.
• The trigonometric functions $\sin x$ and $\cos x$ are continuous at every real number; $\tan x$, $\sec x$, $\cot x$ and $\csc x$ are continuous at every point of their respective domains (excluding the points where they are undefined).

How to use these results. To decide continuity of a given function, first recognise it as one of these standard types (or as a combination of them, using the algebra rules of §5). A polynomial needs no testing — it is continuous everywhere. A rational function needs only its denominator's zeros located; it is continuous everywhere else.

Illustration. For $f(x) = \dfrac{x+1}{x^2+1}$, the denominator $x^2 + 1$ is never zero for any real $x$ (since $x^2 \ge 0$ gives $x^2 + 1 \ge 1 > 0$). Being a rational function with a nowhere-zero denominator, $f$ is therefore continuous at every real number. …