Algebra of continous functions
Algebra of continous functions
Last updated at May 6, 2021 by Teachoo
Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let π(π)=ππ¨π¬β‘π To check continuity of π(π₯), We check itβs if it is continuous at any point x = c Let c be any real number f is continuous at π₯ =π if if L.H.L = R.H.L = π(π) i.e. limβ¬(xβπ^β ) π(π₯)= limβ¬(xβπ^+ ) " " π(π₯)= π(π) LHL at x β c limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) cosβ‘(πββ) = limβ¬(hβ0) sinβ‘π sinβ‘β+cosβ‘π cosβ‘β Putting β=0 = sinβ‘π sinβ‘0+cosβ‘π cosβ‘0 = 0+cos c . 1 = πππβ‘π π΄π , cosβ‘(π₯βπ¦) =cosβ‘π₯ cosβ‘π¦βsinβ‘π₯ sinβ‘π¦ RHL at x β c limβ¬(xβπ^+ ) f(x) = limβ¬(hβ0) f(c + h) = limβ¬(hβ0) cosβ‘(π+β) = limβ¬(hβ0) cosβ‘π cosβ‘β β sinβ‘π sinβ‘β Putting β=0 = cosβ‘π cosβ‘0 β sinβ‘π sinβ‘0 = cos c . 1 β 0 = πππβ‘π π΄π , cosβ‘(π₯+π¦) =cosβ‘π₯ cosβ‘π¦βsinβ‘π₯ sinβ‘π¦ And, π(π) = cosβ‘π Since, L.H.L = R.H.L = π(π) β΄ Function is continuous at x = c Thus, we can write that f is continuous for x = c , where c βπ β΄ πππβ‘π is continuous for every real number.