Continuity of fractional part function for negative integers

Question

Given a function $f(x)=frac(x^2)-(frac(x))^2$

We have to check continuity at $x=-2$

I know that $frac(x)= x-[x]$ where $[x]$ is the greatest integer function less than or equal to x.

Checking continuity -

$f(-2)=0$

$f(-2^+)=frac(4^+)-(frac(-2^+))^2=0$

But according to the textbook, right-hand limit should be 1. And hence the function should be discontinuous. Where am I wrong? I used the same way to check continuity at $x=2$ and it came out to be continuous which is correct according to the textbook.

Answer 1

To check continuity you want to evaluate the function close to (say, just above) $-2$. If $x$ is just above $-2$ then $x^2$ is just below $4$. So $\mathrm{frac}(x^2)$ is a number just below $1$.