Given function \(f: R \rightarrow R\), such that
\(f(x)=\left[\begin{array}{cl}
5, & \text { if } x \leq 1 \\
a+b x, & \text { if } 1 < x < 3 \\
b+5 x, & \text { if } 3 \leq x < 5 \\
30, & \text { if } x \geq 5
\end{array}\right.\)
If \(f\) is continuous at \(x=1\), then
\(a+b=5\) ...(i)
If \(f\) is continuous at \(x=3\), then
\(\Rightarrow \quad \begin{gathered}
a+3 b=b+15 \\
a+2 b=15 \quad \ldots (ii)
\end{gathered}\)
and if \(f\) is continuous at \(x=5\), then
\(b+25=30 \Rightarrow b=5\) ...(iii)
From Eqs. (ii) and (iii), we get
\(a=5\) ...(iv)
but \(a=5\) and \(b=5\) doesn't satisfy the Eq. (i). so, \(f: R \rightarrow R\) is not continuous for any values of \(a\) and \(b\).
Hence, option (4) is correct.