Check sibling questions

Ex 5.1, 19 - Ex 5.1

This video is only available for Teachoo black users

Ex 5.1, 19 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

Ex 5.1, 19 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Ex 5.1, 19 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

This video is only available for Teachoo black users

Get live Maths 1-on-1 Classs - Class 6 to 12


Transcript

Ex 5.1, 19 (Introduction) Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Ex 5.1, 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Given g(x) = x − [𝑥] Let c be an integer g(x) is continuous at 𝑥 =𝑐 if L.H.L = R.H.L = 𝑔(𝑐) if lim┬(x→𝑐^− ) 𝑔(𝑥)=lim┬(x→𝑐^+ ) " " 𝑔(𝑥)=𝑔(𝑐) LHL at x → c lim┬(x→𝑐^− ) g(x) = lim┬(h→0) g(c − h) = lim┬(h→0) (𝑐−ℎ)−[𝒄−𝒉] = lim┬(h→0) (𝑐−ℎ)−(𝒄−𝟏) = lim┬(h→0) 𝑐−ℎ−𝑐+1 = lim┬(h→0) −ℎ+1 = 0+1 = 1 RHL at x → c lim┬(x→𝑐^+ ) g(x) = lim┬(h→0) g(c + h) = lim┬(h→0) (𝑐+ℎ)−[𝒄+𝒉] = lim┬(h→0) (𝑐−ℎ)−(𝒄) = lim┬(h→0) −ℎ = 𝟎 Since LHL ≠ RHL ∴ g(x) is not continuous at x = c Hence, g(x) is not continuous at all integral points. Hence proved

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.