



Last updated at Dec. 8, 2016 by Teachoo
Transcript
Ex 5.1, 8 Find all points of discontinuity of f, where f is defined by 𝑓𝑥=𝑥𝑥, 𝑖𝑓 𝑥≠0&0 , 𝑖𝑓 𝑥=0 Given 𝑓𝑥=𝑥𝑥, 𝑖𝑓 𝑥≠0&0 , 𝑖𝑓 𝑥=0 Case 1 At x = 0 f is continuous at x = 0 if L.H.L = R.H.L = 𝑓0 i.e. limx→0− 𝑓𝑥 = limx→0+ 𝑓𝑥 = 𝑓0 Since, L.H.L ≠ R.H.L ⇒ f is discontinuous at 𝑥 =0 Case 2 Let x = c (where c > 0) ∴ 𝑓𝑥=𝑥𝑥 = 𝑥𝑥 = 1 f is continuous at x = c if limx→𝑐 𝑓𝑥=𝑓(𝑐) Thus limx→𝑐 𝑓𝑥=𝑓(𝑐) ⇒ f is continuous for 𝑥 =𝑐 greater than 0. ⇒ f is at continuous for all real numbers greater than 0. Case 3 Let x = c (where c < 0) 𝑓𝑥= 𝑥𝑥 𝑓𝑥=−𝑥𝑥 𝑓𝑥= − 1 f is continuous at x = c if limx→𝑐 𝑓𝑥=𝑓(𝑐) Thus , limx→𝑐 𝑓𝑥=𝑓(𝑐) ⇒ f is continuous for 𝑥 =𝑐 ( c is less than 0 ) ⇒ f is continuous for all real numbers less than 0. Hence, only x=0 is point is discontinuity. ⇒ f is continuous for all real numbers except 0. Thus, f is continuous for x ∈ R − {0}
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