If $f\left(x\right)=\left[x\right]$, (where $\left[.\right]$ represents greatest integer function) for $x\in \left(-1,2\right)$, then $f$ is discontinuous at

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=\frac{x^2-3x-10}{x-5}$, for $3\le x\le 6, x\ne 5$

$=10$, for $x=5$

$=\frac{x^2-3x-10}{x-5}$, for $6<x\le 9$.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=2x^2-2x+5$, for $0\le x\le 2$

$=\frac{1-3x-x^2}{1-x}$, for $2<x<4$

$=\frac{x^2-25}{x-5}$, for $4\le x\le 7$ and $x\ne 5$

$=7$, for $x=5$.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=\frac{\cos 4x-\cos 9x}{1-\cos x}$, for $x\ne 0$

$f\left(0\right)=\frac{68}{15}$, at $x=0$ on $-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}$.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=\frac{{\sin }^2\mathrm{\pi }x}{3{\left(1-x\right)}^2}$, for $x\ne 1$

$=\frac{{\mathrm{\pi }}^2{\sin }^2\left(\frac{\mathrm{\pi }x}{2}\right)}{3+4{\cos }^2\left(\frac{\mathrm{\pi }x}{2}\right)}$, for $x=1$, at $x=1$.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=\frac{\left|x+1\right|}{2x^2+x-1}$, for $x\ne -1$

$=0$, for $x=-1$ at $x=-1$.

Discuss the continuity of the following functions at the point(s) or on the interval indicated against them.

$f\left(x\right)=\left[x+1\right]$, for $x\in [-2, 2)$

Where $\left[.\right]$ is greatest integer function.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=2x^2+x+1$, for $\left|x-3\right|\ge 2$

       $=x^2+3$,      for $1<x<5$.

Identify discontinuities if any for the following function as either a jump or a removable discontinuity on their respective domains.

$f\left(x\right)=x^2+x-3$, for $x\in [-5,-2)$

        $=x^2-5$,    for $x\in (-2,5]$.