Question

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

Answer 1

`" 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.

\[\therefore f\left( x \right) = hog\left( x \right) = h\left( g\left( x \right) \right) = h\left( \left| x \right| \right) = \sin \left| x \right|\]