If $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\left|\mathrm{x}\right|-3, \mathrm{x}<1\\ \left|\mathrm{x}-2\right|+\mathrm{a}, \mathrm{x}\ge 1\end{array}\right. \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)=\left\{\begin{array}{l}2-\left|\mathrm{x}\right|, \mathrm{x}<2\\ \mathrm{sgn}\left(\mathrm{x}\right)-\mathrm{b}, \mathrm{x}\ge 2\end{array}\right.$and  $\mathrm{h}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)$ is discontinuous at exactly one point, then which of the following values of a and b are possible?

f (x) is continuous for all x if it is continuous at  x = 1,

$li{m}_{x\to 1^{-}}\left(f\right(x\left)\right)=li{m}_{x\to 1^{+}}\left(f\right(x\left)\right)$

 $$\begin{gathered} \left|1\right|-3=\left|1-2\right|+\mathrm{a} \\ \Rightarrow \mathrm{a}=-3 \end{gathered}$$

g(x) is continuous for all x if it is continuous at x = 2,

for which $2-\left|2\right|=\mathrm{sgn}\left(2\right)-\mathrm{b}=1-\mathrm{b} \mathrm{or} \mathrm{b}=1.$

Thus, f(x) + g(x) is continuous for all x if $\mathrm{a}=-3, \mathrm{b}=1$

Hence, f(x) is discontinuous at exactly one point for options (1) and (2).