The sum of values of a and b such that the function $f(x)$ defined by $f(x)=\left\{\begin{matrix}3,&x≤1\\ax+b,&1<x<5\\10,&x≥5\end{matrix}\right.$ is a continuous function is

3

$f(x)=\left\{\begin{matrix}3,&x≤1\\ax+b,&1<x<5\\10,&x≥5\end{matrix}\right.$

$f(1)=3$

$\underset{x→1}{\lim}=a+b$

$⇒a+b=3$   ...(1)

$f(5)=10$

$\underset{x→5}{\lim}=5a+b$

$⇒5a+b=10$   ...(2)

Subtracting eq. (1) from eq. (2)

$4a=7⇒a=\frac{7}{4}$ from (1) $b=3-\frac{7}{4}=\frac{5}{4}$

$a+b=3$