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Find the Point of Discontinuity, If Any, of the Following Function: F ( X ) = { Sin X − Cos X , If X ≠ 0 − 1 , If X = 0 - Mathematics

Sum

Find the point of discontinuity, if any, of the following function: \[f\left( x \right) = \begin{cases}\sin x - \cos x , & \text{ if }  x \neq 0 \\ - 1 , & \text{ if }  x = 0\end{cases}\]

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Solution

The given function f is \[f\left( x \right) = \begin{cases}\sin x - \cos x , & \text{ if }  x \neq 0 \\ - 1 , & \text{ if }  x = 0\end{cases}\]

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

`" if c ≠ 0 , then " f ( c) = sin c - cos c `

`lim_(x → c)  f ( x) = lim_ ( x→c )  ( sin x - cos x ) = sin c - cos c `

`∴ lim _ (x →c) f ( x) = f ( c) `

Therefore, f is continuous at all points x, such that x ≠ 0

Case II:

 if c = 0 , then f (0) = - 1

`lim _ (x →0^-) f(x) = lim _ (x →0^-)(sin x - cos x ) = sin 0 - cos 0 = 0- 1 =- 1`

`lim _ (x →0^ +) f (x) = lim _ (x →0)(sin x - cos x ) = sin 0 -  cos 0 = 0 - 1 =  - 1`

`∴ lim _ (x →0^-) f (x) = lim _ (x →0^+) f (x)= f(0) `

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.2 | Q 3.12 | Page 34
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