Let ,f:[-1,2] R be given by f(x)=2x^2+x+[x^2]-[x], where [t] denotes the greatest integer …

Question

Let ,$f:[-1,2]	o R$ be given by $f(x)=2{x}^{2}+x+[{x}^{2}]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

Accepted Answer

$$\begin{aligned}
& f(x)=2 x^2+x+\left[x^2\right]-[x]=2 x^2+\left[x^2\right]+\{x\} \
& f(-1)=2+1+0=3 \
& f\left(-1^{+}\right)=2+0+0=2 \
& f\left(0^{-}\right)=0+1=1 \
& f\left(0^{+}\right)=0+0+0=0 \
& f\left(1^{+}\right)=2+1+0=3
\end{aligned}$$

$$\begin{aligned}
& f\left(1^{-}\right)=2+0+1=3 \
& f\left(2^{-}\right)=8+3+1=12 \
& f\left(2^{+}\right)=8+4+0=12
\end{aligned}$$

$$\therefore$$ discontinuous at $$x=0, \sqrt{2}, \sqrt{3},-1$$

Solutions · 1