Let c be in S subset R and f: S->R be a function. Assume that for every sequence

Question

Let c be in S subset R and f: S->R be a function. Assume that for every sequence {Xn} subset S with the property that lim Xn=c the sequence {f(Xn)} is convergent. Show that f is continuous at c.

Explanation / Answer

a) f(x) = (x-1) / (x+1) at c= -1 lim x-->-1 (x-1)/(x+1) = (-1-1)/(-1+1) = -2/0 (undefined) f(x) is not continuous at x=-1 b) f(x) = x^2+2 lim x-->4 (x^2+2) = 4^2+2 = 18 f(4) = 18 f(x) is continuous at x=4 c) Examine the limit from below 2 lim x-->2- f(x) = lim x--> 2- (x-1) = (2-1) = 1 Examine the limit from above 2 lim x-->2+ f(x) = lim x-->2+ 2x = 4 f(2) = 2 lim x-->2- is not equal to lim x-->2+ f(x) Therefore, f(x) is not continuous at x=2 since the limit does not exist at x=2

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