Solveeit Logo
Login

Question

Question: The function ƒ(x) =\(\frac{1}{3}\) is –...

The function ƒ(x) =13\frac{1}{3} is –

A

Discontinuous at only one point

B

Discontinuous exactly at two points

C

Continuous everywhere

D

None of these

Answer

Discontinuous at only one point

Explanation

Solution

The only doubtful point is x = 0.

LHL = ƒ(0 – h) = lim⁡h→0\lim _ { h \rightarrow 0 } (−h+1)2−(1 h−1 h)( - \mathrm { h } + 1 ) ^ { 2 - \left( \frac { 1 } { \mathrm {~h} } - \frac { 1 } { \mathrm {~h} } \right) }

= lim⁡h→0\lim _ { h \rightarrow 0 } (1 – h)2 = 1

RHL = lim⁡h→0\lim _ { h \rightarrow 0 } ƒ(0 + h) = lim⁡h→0\lim _ { h \rightarrow 0 } (h+1)2−(1 h+1 h)( \mathrm { h } + 1 ) ^ { 2 - \left( \frac { 1 } { \mathrm {~h} } + \frac { 1 } { \mathrm {~h} } \right) }

= (1+h)2−2h( 1 + h ) ^ { 2 - \frac { 2 } { h } } =(1 + h)2 [(1 + h)1/h]–2

Since LHL ≠ RHL,

∴ ƒ(x) is not continuous at x = 0.