$f\left(x\right)=\left\{\begin{array}{c}5:x\le 1\\ a+bx:1<x<3\\ b+5x:3\le x<5\\ 30:x\ge 5\end{array}\right.$

For $f\left(x\right)$ to be continuous at $x=5$

$\underset{x\to 5^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=\underset{x\to 5^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=f\left(5\right)$

$\Rightarrow \underset{x\to 5^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(b+5x\right)=30$

$\Rightarrow b+25=30\Rightarrow b=5$

For $f\left(x\right)$ to be continuous at $x=1$

$\underset{x\to 1^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=\underset{x\to 1^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)=f\left(1\right)$

$\Rightarrow \underset{x\to 1^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(a+5x\right)=5$

$\Rightarrow a=0$

Now at $x=3,$

$LHL=\underset{x\to 3^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(0+5x\right)=15$

$RHL=\underset{x\to 3^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(5+5x\right)=20$

Hence function $f\left(x\right)$ is discontinuous at $x=3$ for all $a,b\in R$ .