For any $y \in \mathbb R$, let $[y]$ denotes the greatest integer less than or equal to $y$. Define $f: \mathbb R^2 \to \mathbb R$ be $f(x,y) = x^{[y]}$. Then there are four options from which any number of option can be correct.
But my question is how can we define this function at $(0,0)$?
Can anyone give any suggestion?
Edit:** I know the function is not defined on whole $R^2$ . But This question was asked in a National Level Exam in India. That's why I had doubt in myself.**
the problem should have included a value for $f(0,0)$. If $|y|<1$, then $$ [y]=\begin{cases}\phantom{-}0 & \text{if }y\ge0,\\-1 &\text{if }y<0.\end{cases} $$ Then, if $x\ne0$, $$ x^{[y]}=\begin{cases}1 & \text{if }y\ge0,\\\dfrac{1}{x} &\text{if }y<0.\end{cases} $$ $f$ cannot be defined at $(0,0)$ to make it continuous. However, answer 2 is correct.