Are the graphs of sec and csc in continuous line?

Question

We were going over the graph of secant and cosecant and I noticed that if I did $y=\sec x$ the lines would come into view and then go back out of view even if I zoomed out on my calculator. I've always thought that a graph of a line would only be one line, but is that not the case for secant and cosecant? Does the graph have one line or multiple lines?

Answer 1

A graph does not have to consist of one connected line. Consider the function: $$f(x) = \begin{cases} 1 & x \ge 0 \\ 0 & x < 0 \end{cases}$$

This graph is made of two separate lines.

Similarly, consider $g(x) = 1/x$. There are two separate curves on this graph; they don't connect at $0$.

A graph can be even more messed up! Consider $h(x) = 1$ if $x$ is rational, and $0$ if $x$ is irrational. Drawing this is quite the challenge.

In general, you might want to look into the concept of continuity.