Discuss the continuity of f(x)= |x|

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Question

Discuss the continuity of $\text{f(x)}=\sin|\text{x}|$

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Answer

Let $\text{f(x)}=\sin|\text{x}|$

This function f is defined for every real number and f can be written as the composition of two functions as,

f = goh, where g(x) = |x| and $\text{h(x)}=\sin\text{x}$

$\big[\because (\text{goh})(\text{x})=\text{g(h(x))}=\text{g}(\sin\text{x})=|\sin\text{x}|=\text{f(x)}]\Big]$

It has to be proved first that g(x) = |x| and $\text{h(x)}=\sin\text{x}$ are continuous functions.

g(x) = |x| can be written as

$\text{g(x)}=\begin{cases}-\text{x},&\text{if }\text{ x}<0\\\text{x},&\text{if }\text{ x}\geq0\end{cases}$

Clearly, g is defined for all real numbers.

Let c be real number.

Case I:

If c < 0, then g(c) = -c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(-\text{x})=-\text{c}$

$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$

Therefore, g is continuous at all points x, such that x < 0

Case II:

If c > 0, then g(c) = c and $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\text{x})=\text{c}$

$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$

Therefore, g is continuous at all points x, such that x > 0

Case III:

If c = 0, then g(c) = g(0) = 0

$\lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^-}\text{g}(-\text{x})=0$

$\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=0$

$\therefore\ \lim\limits_{\text{x}\rightarrow0^-}\text{g(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{g(x)}=\text{g}(0)$

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

$\text{h(x)}=\sin\text{x}$

It is evident that $\text{h(x)}=\sin\text{x}$ is defined for every real number.

Let c be a real number. Put x = c + k

If x → c, then k → 0

$\text{h(c)}=\sin\text{c}$

$\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin\text{x}$

$=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin(\text{c}+\text{k})$

$=\lim\limits_{{\text{x}}\rightarrow\text{c}}\big[\sin\text{c}\cos\text{k}+\cos\text{c}\sin\text{k}\big]$

$=\lim\limits_{{\text{x}}\rightarrow\text{c}}(\sin\text{c}\cos\text{k})+\lim\limits_{{\text{x}}\rightarrow\text{c}}(\cos\text{c}\sin\text{k})$

$=\sin\text{c}\cos0+\cos\text{c}\sin0$

$=\sin\text{c}+0$

$=\sin\text{c}$

$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\text{g(c)}$

Therefore, h is a continuous function.

It is know that for real valued functions g and h, such that (goh) is defind at c, if g is continuse at c and if f is continuous at g(c), then (fog) is continuous at c.

Therefore, $\text{f(x)}=(\text{goh})(\text{x})=\text{g(h(x))}=\text{g}(\sin\text{x})=|\sin\text{x}|$ is a continuous function.