Algebra of continous functions
Algebra of continous functions
Last updated at April 16, 2024 by Teachoo
Ex 5.1, 19 (Introduction) Show that the function defined by g (x) = x β [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Ex 5.1, 19 Show that the function defined by g (x) = x β [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Given g(x) = x β [π₯] Let c be an integer g(x) is continuous at π₯ =π if L.H.L = R.H.L = π(π) if limβ¬(xβπ^β ) π(π₯)=limβ¬(xβπ^+ ) " " π(π₯)=π(π) LHL at x β c limβ¬(xβπ^β ) g(x) = limβ¬(hβ0) g(c β h) = limβ¬(hβ0) (πββ)β[πβπ] = limβ¬(hβ0) (πββ)β(πβπ) = limβ¬(hβ0) πβββπ+1 = limβ¬(hβ0) ββ+1 = 0+1 = 1 RHL at x β c limβ¬(xβπ^+ ) g(x) = limβ¬(hβ0) g(c + h) = limβ¬(hβ0) (π+β)β[π+π] = limβ¬(hβ0) (πββ)β(π) = limβ¬(hβ0) ββ = π Since LHL β RHL β΄ g(x) is not continuous at x = c Hence, g(x) is not continuous at all integral points. Hence proved