Why absolute-value function is not a continuous function
I am doing a pre-calculus course from:
It tells me that a continuous function has continuous graphs while a piece-wise function has kind of step-based graph. Then it tells that absolute value function |x| is a piece-wise function. I noticed the |x| has a continuous graph without any breaks. How come it is defined piece-wise then ?
Wikipedia says it is both continuous and piece-wise linear function. I am confused:
$\endgroup$ 12 Answers
$\begingroup$You are confusing "piecewise" with "step". The absolute value function has a piecewise definition, but as you and the text correctly observe, it is continuous. Informally, the pieces touch at the transition points.
The greatest integer function has a piecewise definition and is a step function. There are breaks in its graph at the integers.
I think you should read the text carefully to see whether it is as confused as you are. If so, then it should be corrected.
$\endgroup$ 3 $\begingroup$Imagine you have a piecewise function $$f(x)=\left\{\begin{matrix}g(x) & x\leq a \\ h(x) & x>a\end{matrix}\right.$$ $f(x)$ is continuous if $g(x)$ is continuous when $x\leq a$, $h(x)$ is continuous when $x>a$, and $g(a)=h(a)$ (which is indeed the case for the absolute value function).
$\endgroup$
