Question

Let $g$ be the greatest integer function. Then the function $f\left(x\right)=\left(g\right(x){)}^2-g(x)$ is discontinuous at

• A. All integers
• B. All integers except $0$ and $1$
• C. All integers except $0$
• D. All integers except $1$

Answer 1

The correct option is A All integers

Function is continious at k if $\underset{x\to k}{lim}f\left(x\right)$ exist and $\underset{x\to k}{lim}f\left(x\right)=f\left(k\right)$

given g(x) is greatest integer funtion $f\left(x\right)=\left(g\right(x){)}^2-g(x)$

Lets check its continiuty at integer k

$\underset{x\to k-}{lim}\left(g\right(x){)}^2-g(x)=\underset{h\to 0}{lim}\left(g\right(k-h){)}^2-g(k-h)={(k-1)}^2-(k-1)$

$\underset{x\to k+}{lim}\left(g\right(x){)}^2-g(x)=\underset{h\to 0}{lim}\left(g\right(k+h){)}^2-g(k+h)={\left(k\right)}^2-\left(k\right)$

We can see that LHL$\ne$ RHL

So this function will be discontinuous at all integers.