Let $f(x) = [x^3-3]$ where x is the greatest integer function. Then find the number of points in the interval (1,2) where this function is discontinuous ?
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[Math] Finding the points of disctontinuity of greatest integer function
calculusfunctions
Best Answer
HINT: I’m going to use the more modern notation $\lfloor x\rfloor$ for the greatest integer (or floor) function instead of $[x]$. We’re interested in points of discontinuity of the function $f(x)=\lfloor x^3-3\rfloor$ on the interval $(1,2)$. The floor function is constant on intervals between consecutive integers and jumps at each integer, so it has a discontinuity at each integer. Thus, $f(x)$ will have a discontinuity at each $x\in(1,2)$ at which $x^3-3$ is an integer. So for what real numbers $x\in(1,2)$ is it true that $x^3-3$ is an integer?