The question
$f(x)=[2x+\sqrt n]$, where $[x]$ is the greatest integer less than or equal to $x$ and $n<100$. If $f(x)$ is discontinuous in the interval $[1,1.5)$, then find the total number of values of $n$.
I tried by putting some values of $n$, but did not find any pattern of their discontinuity in the given interval. I tried to visualize the problem using Desmos, and found that there are many values of $n$ which satisfy this condition. But I do not know how to approach the problem in a proper manner.
Thanks in advance.
Start by writing inequality on x given by domain definition Multiply x by 2 and add root n while simultaneously performing same operations across inequality Proceed by putting different values of n ( only on the outside part of inequality for better understanding) and then evaluate what the GIF would be depending on this inequality. Hopefully you will see the pattern of discontinuity at each perfect square)