Browse Questions

# Show that the function defined by f (x) = | cos 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 $g(x)=|x|$
$h(x)=\cos x$
$f(x)=(goh)(x)$
$\quad\;\;\;=g(h(x))$
$\quad\;\;\;=g(\cos x)$
$\quad\;\;\;=\mid \cos x\mid$
$g(x)=|x|$ and $h(x)=\cos x$
Both are continuous for all values of $x\in R$
Step 2:
$(goh)(x)$ is also continuous.
$f(x)=(goh)(x)$
$\quad\;\;\;=|\cos x|$
$f(x)=|\cos x|$ is continuous for all values of $x\in R$