AIEEE 2012 (Offline) 29th April Code AQuestion 1
If $f:\mathbf{R}\to\mathbf{R}$ is a function defined by $f(x)=[x]\cos\left(\frac{2x-1}{2}\right)\pi$, where $[x]$ denotes the greatest integer function, then $f$ is :
Limits & Continuity
Solution
$f(x)=[x]\sin\pi x$. Away from integers it is continuous. At an integer $n$: $[x]$ is bounded near $n$ and $\sin\pi x\to0=\sin\pi n$, so both one-sided limits equal $f(n)=0$. Hence $f$ is continuous for every real $x$.