Let $g(x) = x^2 + [x^2]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which

Question

Let $g(x) = x^2 + [x^2]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which of the following statements is/are CORRECT?

Professor PC Thomas Classes · Answer

The function is discontinuous where $x^2$ is a positive integer, i.e., $x = \pm \sqrt{k}$ for $k \in \mathbb{N}$. Let us check $x=0$: as $x 	o 0$, $x^2 	o 0^+$, so $[x^2] = 0$. The limit is $0+0=0$, and $g(0)=0$, so it is continuous at $x=0$, making (A) correct. At $x=\sqrt{k}$ (like $\sqrt{2}$), the left limit has $[x^2] = k-1$ and the right limit has $[x^2] = k$, leading to a jump discontinuity, making (B) and (C) correct. For (D), as $x 	o 1^+$, $x^2 	o 1^+$, so $[x^2] = 1$. The limit is $1^2 + 1 = 2$. Thus, (D) is also correct.