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 $\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{+}}{lim}f(x)=\underset{x\to a}{lim}f(x)$

Calculation

Doubtful points are x = n, n∈I

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

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

⇒  f(x) = 0 ∀ x ∈ R

∴ f is continuous for every real x.

The correct answer is Option 4.