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Question
Discuss the continuity of f(x) = sin | x |.
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Solution
`" Let " f(x)=sin |x| `
This function f is defined for every real number and f can be written as the composition of two functions as, f = h o g, where g (x) = |x| and h (x) = sin x
\[\left[ \because hog\left( x \right) = h\left( g\left( x \right) \right) = h\left( \left| x \right| \right) = \sin \left| x \right| \right]\]
It has to be proved first that `g(x)=|x|` and `h(x)=sinx` are continous function .
`g(x)=|x|`can be written as
`g(x)=[[-x, if x < 0],[x, if x≥ 0]]`
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
`if c < 0 " then " g(c)=-c ` and `lim_(x->c)g(x)=lim_(x->c)(-x)=-c`
`∴lim_(x->c) g (x) = g(c)`
Therefore, g is continuous at all points x > 0
Case III:
` " If " c=0, `then `g(c)=g(0)=0`
`lim_(x->0)g(x)=lim_(x->0)(x)=0`
`lim_(x->0)g(x)=lim_(x->0)(x)=0`
`∴lim_(x->0)g(x)=lim(x)=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.
Now, h (x) = sin x
It is evident that h (x) = sin x is defined for every real number.
Let c be a real number.
Put x = c + k
If x → c, then k → 0
h (c) = sin c
`h(c)=sin c`
`lim_(x->c)(x)=lim_(x->c)sin x`
`=lim_(k->0)sin(c+k)`
`=lim_(k->0)[sin c cos k + cos c sin k]`
`= lim_(k->0)(sin c cos k)+lim_(k->0)(cos c sin k)`
`=sin c cos 0 + cos c sin 0`
`=sin c+0`
`=sin c`
`∵lim_(x->c)h(x)=g(c)`
So, h is a continuous function.
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