# Misc 21

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 21 Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer. Consider the function 𝑓 𝑥= 𝑥+ 𝑥−1 𝑓 is continuous every where , but it is not differentiable at 𝑥 = 0 & 𝑥 = 1 . 𝑓 𝑥= −𝑥− 𝑥−1 𝑥≤0𝑥− 𝑥−1 0<𝑥<1𝑥+ 𝑥−1 𝑥≥1 = −2𝑥+1 𝑥≤0 1 0<𝑥<12𝑥−1 𝑥≥1 Checking continuity Case 1 :- At 𝑥<0 𝑓 𝑥=−2𝑥+1 𝑓 𝑥 is polynomial ⇒ 𝑓 𝑥 is continuous Case 2 :- At 𝑥>1 𝑓 𝑥=2𝑥−1 𝑓 𝑥 is polynomial ⇒ 𝑓 𝑥 is continuous Case 3 :- At 0<𝑥<1 𝑓 𝑥=1 𝑓 𝑥 is constant, ⇒ 𝑓 𝑥 is continuous Case 4 :- At 𝑥=0 𝑓 𝑥 = −2𝑥+1 𝑥≤0 1 0<𝑥<12𝑥−1 𝑥≥1 A function is continuous at 𝑥=0 if LHL = RHL = 𝑓 0 i.e. lim𝑥 → 0− 𝑓 𝑥 = lim𝑥 → 0+ 𝑓 𝑥 = 𝑓 0 & 𝑓 𝑥= −2𝑥+1 𝑓 0= −2 0+1= 1 Hence LHL = RHL = f (0) ⇒ 𝑓 is continuous . Case 5 :- At 𝑥=1 A function is continuous at 𝑥=1 if LHL = RHL = 𝑓 1 i.e. lim𝑥 → 1− 𝑓 𝑥 = lim𝑥 → 1+ 𝑓 𝑥 = 𝑓 1 & 𝑓 𝑥=2𝑥−1 𝑓 1=2 1−1=2−1 =1 Hence LHL = RHL = 𝑓 1 ⇒ 𝑓 is continuous at 𝑥=1 Thus 𝑓 𝑥= 𝑥+ 𝑥−1 is continuous for all value of 𝑥 . To check differentiability Case 1 :- At 𝑥<1 𝑓 𝑥=−2𝑥+1 𝑓 𝑥 is polynomial ⇒ 𝑓 𝑥 is differentiable Case 2 :- At 𝑥>1 𝑓 𝑥=2𝑥−1 𝑓 𝑥 is polynomial ⇒ 𝑓 𝑥 is differentiable Case 3 :- At 0<𝑥<1 𝑓 𝑥=1 𝑓 𝑥 is constant ⇒ 𝑓 𝑥 is differentiable Case 4 :- At 𝑥=0 We know that 𝑓 is differentiate at 𝑥 = 0 If L.H.D = R.H.D = 𝑓′ 0 i.e., limℎ → 0− 𝑓 0 − 𝑓 0 − ℎℎ = limℎ → 0+ 𝑓 0 + ℎ − 𝑓 0ℎ = 𝑓′ 𝑐

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.