If f : R → R is a function defined by f(x) = [x – 1] cos ((2x – 1)/2)π, where [.] denotes the greatest integer function, then f is:

Concept:

A function f(x) is continuous at x = a if \(\lim_{x\to a^-}\ f(x)=\lim_{x\to a^+}\ f(x)=\lim_{x\to a}\ f(x)\)

Calculation

Doubtful points are x = n, n∈I

LHL = \(\lim _{x \rightarrow n^{-}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-2) \cos \left(\frac{2 n-1}{2}\right) \pi=0\)

RHL = \(\lim _{x \rightarrow n^{+}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-1) \cos \left(\frac{2 n-1}{2}\right) \pi=0\)

⇒  f(x) = 0 ∀ x ∈ R

∴ f is continuous for every real x.

The correct answer is Option 4.