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Example 3 Discuss the continuity of the function f given by 𝑓(π‘₯) =|π‘₯| π‘Žπ‘‘ π‘₯ = 0. 𝑓(π‘₯) = |π‘₯| 𝑓(π‘₯)= {β–ˆ(βˆ’π‘₯, 𝑖𝑓 π‘₯<0@π‘₯, 𝑖𝑓 π‘₯ β‰₯0)─ f is continuous at π‘₯ = 0 if L.H.L = R.H.L = 𝑓(0) i.e. (π‘™π‘–π‘š)┬(π‘₯β†’0^βˆ’ ) 𝑓(π‘₯)=(π‘™π‘–π‘š)┬(π‘₯β†’0^+ ) 𝑓(π‘₯)=𝑓(0) Finding LHL and 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| = lim┬(hβ†’0) h = 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| = lim┬(hβ†’0) h = 0 And, f(0) = 0 So, LHL = RHL = f(0) Hence, f is continuous at 𝒙 = 𝟎

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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.