If \(f(x)=\left\{\begin{array}{cc}\frac{x-3}{|x-3|}+a & , x<3 \\ a+b & , x=3 \\ \left\lvert\, \frac{x-3}{x-3}+b\right. & , x>3\end{array}\right.\)
Is continuous at \(x=3\), then the value of \(\mathrm{a}-\mathrm{b}\) is

If \(x < 3\), then
\(\frac{x-3}{|x-3|}+\mathrm{a}=\frac{x-3}{-(x-3)}+\mathrm{a}=\mathrm{a}-1\)
If \(x>3\), then \(\frac{|x-3|}{x-3}+\mathrm{b}=\frac{x-3}{x-3}+\mathrm{b}=1+\mathrm{b}\)
\(\therefore \quad\) Given function can be written as
\(\mathrm{f}(x)=\left\{\begin{array}{l}
\mathrm{a}-1, x < 3 \\
\mathrm{a}+\mathrm{b}, x=3 \\
1+\mathrm{b}, x>3
\end{array}\right.\)
As \(\mathrm{f}(x)\) is continuous at \(x=3\), we get
\(\begin{array}{ll}
& \lim _{x \rightarrow 3^{-}} \mathrm{f}(x)=\mathrm{f}(3) \text { and } \lim _{x \rightarrow 3^{+}} \mathrm{f}(x)=\mathrm{f}(3) \\
\therefore \quad & \mathrm{a}-1=\mathrm{a}+\mathrm{b} \quad \text { and } 1+\mathrm{b}=\mathrm{a}+\mathrm{b} \\
\therefore \quad & \mathrm{b}=-1 \quad \text { and } \mathrm{a}=1 \\
\therefore \quad & \mathrm{a}-\mathrm{b}=2
\end{array}\)