Browse Questions

Examine that sin | x | is a continuous function.

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
Let $g(x)=\sin x$,$h(x)=|x|$
$(goh)(x)=g(h(x))$
$\qquad\quad\;\;=g(|x|)$
$\qquad\quad\;\;=\sin |x|$
$\qquad\quad\;\;=f(x)$
Step 2:
$g(x)=\sin x$ and $h(x)=|x|$ both are continuous for all $x\in R$.
$f(x)=(goh)(x)=\sin |x|$ is continuous at all $x\in R$