Solution: The function provided is $f(x)=\left\{\begin{array}{lll}a x+1, & \text { if } & x \leq 3 \\ b x+3, & \text { if } & x>3\end{array}\right.$

Check Continuity at $x=3$,

$\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{-}}(a x+1)=\lim _{h \rightarrow 0}\{a(3-h)+1\}=\lim _{h \rightarrow 0}(3 a-a h+1)=3 a+1$

$\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{+}}(b x+3)=\lim _{h \rightarrow 0}\{b(3+h)+3\}=\lim _{h \rightarrow 0}(3 b+b h+3)=3 b+3$

Also $f(3)=3 a+1$

As a result, $\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)=f(3)$

$\Rightarrow 3 b+3=3 a+1$

$\Rightarrow a-b=\frac{2}{3}$