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Let f(x) be a non-constant continuous function such that lim(x->a) f(x) = lim(x->a) [f(x)], where [.] denotes the greatest integer function and 'a' is a finite real number. Then:

1. lim(x->a) f(x) is an integer
2. lim(x->a) f(x) need not be an integer
3. f(x) has a local minimum at x = a
4. f(x) has a local maximum at x = a

Correct answer: lim(x->a) f(x) is an integer

Solution

Let L = lim(x->a) f(x). Since [.] is continuous at non-integers, lim [f(x)] = [L] if L is non-integer. The condition lim f(x) = lim [f(x)] gives L = [L], which holds only when L is an integer. Hence lim(x->a) f(x) must be an integer.

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