Example 7 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
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Example 7 Is the function defined by f (x) = |x|, a continuous function? f(x) = |π₯| = {β(βπ₯, π₯<0@π₯, π₯β₯0)β€ 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 > 0Case 1 : When x = 0 f(x) is continuous at π₯ =0 if L.H.L = R.H.L = π(0) if limβ¬(xβ0^β ) π(π₯)=limβ¬(xβ0^+ ) " " π(π₯)= π(0) Since there are two different functions on the left & right of 5, we take LHL & RHL . LHL at x β 0 limβ¬(xβ0^β ) f(x) = limβ¬(hβ0) f(0 β h) = limβ¬(hβ0) f(βh) = limβ¬(hβ0) |ββ| = limβ¬(hβ0) β = 0 RHL at x β 0 limβ¬(xβ0^+ ) f(x) = limβ¬(hβ0) f(0 + h) = limβ¬(hβ0) f(h) = limβ¬(hβ0) |β| = limβ¬(hβ0) β = 0 & π(0) = 0 Hence, L.H.L = R.H.L = π(0) β΄ f is continuous at x=0 Case 2 : When x < 0 For x < 0, f(x) = βx Since this a polynomial It is continuous β΄ f(x) is continuous for x < 0Case 3 : When x > 0 For x > 0, f(x) = x Since this a polynomial It is continuous β΄ f(x) is continuous for x > 0 Hence, π(π₯)= |π₯| is continuous at all points. i.e. f is continuous at π β R.
