$f(x) = x[x]$ 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??
Best Answer
You are attempting to evaluate the limit of $f(x)$ as $x\to2$ by substituting $x=2$ and giving the answer $f(2)$. In other words, $$\lim_{x\to a}f(x)=f(a)\ .$$ This is true as long as $f(x)$ is continuous at $x=a$. BUT in this case, that is exactly what the question is asking - is $f(x)$ continuous at $x=2$? So you cannot use this approach.
As suggested in other posts, the best way to do this problem is to look at the left and right hand limits separately. You could try thinking about the value of $[x]$ if $x$ is slightly less than, or slightly greater than $2$: $$[x]=\cases{1&if $1\le x<2$\cr 2&if $2\le x<3$\cr}$$ and therefore $$f(x)=x\,[x]=\cases{x&if $1\le x<2$\cr 2x&if $2\le x<3$.\cr}$$ Hence $$\lim_{x\uparrow2}f(x)=\lim_{x\uparrow2}x=2 \quad\hbox{and}\quad \lim_{x\downarrow2}f(x)=\lim_{x\downarrow2}2x=4\ .$$ Since the left and right hand limits are not equal, $\lim_{x\to2}f(x)$ does not exist. Therefore $f(x)$ is not continuous at $x=2$.