Let [x] denote the greatest integer function.

Question

Let $[x]$ denote the greatest integer function. The number of points in the interval $[2,4]$ at which the function $f(x)=\left[x^2-x-\dfrac12\right]$ is discontinuous, is \_\_\_\_\_\_\_\_.

Answer: 10

Accepted Answer

$f(x)$ is discontinuous wherever $g(x)=x^2-x-\dfrac12$ takes an integer value.

$g(2)=4-2-\dfrac12=\dfrac32,\qquad g(4)=16-4-\dfrac12=\dfrac{23}{2}=11.5$

$g'(x)=2x-1>0$ on $[2,4]$, so $g$ is strictly increasing there, and as $x$ runs over $[2,4]$, $g(x)$ takes every value in $\left[\dfrac32,\dfrac{23}{2}\right]=[1.5,11.5]$ exactly once.

The integers in $[1.5,11.5]$ are $2,3,4,5,6,7,8,9,10,11$ --- that is $10$ integers, each giving exactly one point of discontinuity.

$$\text{Number of points of discontinuity}=10$$

Solutions · 1