Example 13 - Discuss continuity of f(x) = {x, x >= 0 and x^2, x < 0

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Example 13 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Example 13 - Chapter 5 Class 12 Continuity and Differentiability - Part 3 Example 13 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Example 13 Discuss the continuity of the function f given by 𝑓(π‘₯)={β–ˆ(& π‘₯, 𝑖𝑓 π‘₯β‰₯0@& π‘₯2 , 𝑖𝑓 π‘₯<0)─ 𝑓(π‘₯)={β–ˆ(& π‘₯, 𝑖𝑓 π‘₯β‰₯0@& π‘₯2 , 𝑖𝑓 π‘₯<0)─ Since we need to find continuity at of the function We check continuity for different values of x When x = 0 When x < 0 When x > 0 Case 1 : When x = 0 f(x) is continuous at π‘₯ =0 if L.H.L = R.H.L = 𝑓(0) if lim┬(xβ†’0^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’0^+ ) " " 𝑓(π‘₯)= 𝑓(0) Since there are two different functions on the left & right of 0, we take LHL & RHL . LHL at x β†’ 0 lim┬(xβ†’0^βˆ’ ) f(x) = lim┬(hβ†’0) f(0 βˆ’ h) = lim┬(hβ†’0) f(βˆ’h) = lim┬(hβ†’0) (βˆ’h)2 = (βˆ’0)2 = 0 RHL at x β†’ 0 lim┬(xβ†’0^+ ) f(x) = lim┬(hβ†’0) f(0 + h) = lim┬(hβ†’0) f(h) = lim┬(hβ†’0) h = 0 & 𝑓(0) = π‘₯ = 0 Hence, L.H.L = R.H.L = 𝑓(0) ∴ f is continuous at x = 0 Case 2 : When x < 0 For x < 0, f(x) = π‘₯^2 Since this a polynomial It is continuous ∴ f(x) is continuous for x < 0 Case 3 : When x > 0 For x > 0, f(x) = x Since this a polynomial It is continuous ∴ f(x) is continuous for x > 0 Hence, there is no point of discontinuity Thus, f is continuous for all π’™βˆˆπ‘

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.