Examples of Continuous Functions

Rather than testing every new function from the three-condition definition each time, it is convenient to keep a short list of standard families that are already known to be continuous everywhere they are defined, and simply quote them:

(1) A constant function $f(x)=k$ is continuous at every point of $\mathbb{R}$.

(2) Power functions $f(x)=x^n$, with positive integer exponents, are continuous at every point of $\mathbb{R}$.

(3) Polynomial functions $P(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n$ are continuous at every point of $\mathbb{R}$ (they are just sums of constant multiples of power functions).

(4) The trigonometric functions $\sin x$ and $\cos x$ are continuous at every point of $\mathbb{R}$.

(5) The exponential function $a^x$ (for $a>0$) and the logarithmic function $\log_b x$ (for $x>0$, $b>0$, $b\ne1$) are continuous on their domains.

(6) Rational functions, of the form $\dfrac{P(x)}{Q(x)}$ with $Q(x)\ne0$, are continuous at every point $a$ for which $Q(a)\ne0$ — i.e. everywhere except where the denominator vanishes. …