Is the function continuous at the indicated point? $f(x) = x[x]$ at $x=2$ ($[x]$ is the greatest integer function)
I am little confused, it seems like the function does exist at the given point.
when limit goes to 2,
$2[2] = 4$
and $f(2) = 4$
so limit $f(2) = f(2)$
However, when thinking about limit, right-hand side has to be equal to left-hand side. But the function has a jump discontinuity.
so should I say it is continuous at the given point or not??

A function is not continous where it makes a jump because the limit must exist from the left and the right side and these limits must coincide and moreover, the value of the function must be the same as the limits.
