Continuity is the next property of curves that make them nice. Like with limits continuity is defined at a point and we say the function is continuous on an interval if the function is continuous at each point in that interval.
Our first step in understanding continuity is to consider the examples in Figure 1.8.1 which shows examples of points on curves where the function is continuous, and Figure 1.8.2 which shows examples of points on curves where the function is discontinuous.
Based on the example in Figure 1.8.1, Figure 1.8.2, and Figure 1.8.3 describe conditions for continuity. Note it may be easier to describe discontinuity.
From the examples above it may appear the discontinuities occur solely at a single point or on an interval where the function is undefined. However, there exists functions that are discontinuous on an interval where they are defined.
\begin{equation*}
sp(x) = \begin{cases} 0 \amp \text{ if } x \text{ is irrational} \\ 1 \amp \text{ if } x \text{ is rational} \end{cases}
\end{equation*}
\begin{equation*}
confetti(x) = \begin{cases} 1 \amp \text{ if } x \text{ is irrational} \\ \frac{1}{p} \amp \text{ if } x=\frac{n}{p^k} n,k \in \Z^+, p \text{ is prime} \\ 0 \amp \text{ otherwise} \end{cases}
\end{equation*}
