Question
Show that $\text{f(x)}=|\cos\text{x}|$ is a continuous function.
✓
Answer
If f is a real function on a subset of the real numbers and c be a point in the domain off, then f is
continuous at c is $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\text{f(c)}$
Step I:
Let $\text{g(x)}=|\text{x}|$
$\text{h(x)}=\cos\text{x}$
$\text{f(x)}=(\text{goh})(\text{x})$
$=\text{g}(\text{h(x)})$
$=\text{g}(\cos\text{x})$
$={|\cos\text{x}|}$
$\text{g(x)}=|\text{x}|$ and $\text{h(x)}=\cos\text{x}$
Both are continuous for all values of $\text{x}\in\text{R}$
Step II:
(goh)(x) is also continuous
$\text{f(x)}=(\text{goh})(\text{x})$
$={|\cos\text{x}|}$
$\text{f(x)}={|\cos\text{x}|}$ is continuous for all values of $\text{x}\in\text{R}$
continuous at c is $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\text{f(c)}$
Step I:
Let $\text{g(x)}=|\text{x}|$
$\text{h(x)}=\cos\text{x}$
$\text{f(x)}=(\text{goh})(\text{x})$
$=\text{g}(\text{h(x)})$
$=\text{g}(\cos\text{x})$
$={|\cos\text{x}|}$
$\text{g(x)}=|\text{x}|$ and $\text{h(x)}=\cos\text{x}$
Both are continuous for all values of $\text{x}\in\text{R}$
Step II:
(goh)(x) is also continuous
$\text{f(x)}=(\text{goh})(\text{x})$
$={|\cos\text{x}|}$
$\text{f(x)}={|\cos\text{x}|}$ is continuous for all values of $\text{x}\in\text{R}$
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