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Let f(x) = tan(⌊x⌋π) / ⌊1 + |log(sin² x + 1)|⌋, where ⌊·⌋ denotes the greatest integer function and | | denotes the modulus, then f(x) is

1. discontinuous for every x in I
2. continuous for every x
3. non-differentiable for every x in I
4. a periodic function having fundamental period 1

Correct answer: continuous for every x

Solution

The numerator is tan(floor(x)*pi); since floor(x) is an integer n, tan(n*pi)=0 for all x. The denominator floor(1+|log(sin^2 x+1)|) stays equal to 1 because sin^2 x+1 in [1,2] gives log in [0,log2] and 1+that in [1,1.30]. Hence f(x)=0/1=0 for every x, which is continuous everywhere.

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