f(x)={\begin{matrix}5:x\leq 1 \\ a+bx:1<x<3 \\ b+5x:3\leq x<5 \\ 30:x\geq 5\end{matrix}
For f(x) to be continuous at x=5
x→5−limf(x)=x→5+limf(x)=f(5)
⇒x→5−lim(b+5x)=30
⇒b+25=30⇒b=5
For f(x) to be continuous at x=1
x→1+limf(x)=x→1−limf(x)=f(1)
⇒x→1+lim(a+5x)=5
⇒a=0
Now at x=3,
LHL=x→3−lim(0+5x)=15
RHL=x→3+lim(5+5x)=20
Hence function f(x) is discontinuous at x=3 for all a,b∈R .