Multiplication of a continuous function with non continuous $$f(x) = \begin{cases} e^x e^{-1}  &  ,0 \leq x\leq 1
\\   \lfloor x \rfloor+ 1 & ,1< x < 3 \end{cases}$$
 and $$g(x) = x^2 -ax + b $$ such that $f(x)g(x)$ is continuous in $[0,3)$ then find the ordered pair $(a,b)$

You need to make it continuous at $x=1$. That is
\begin{align}
   \lim_{x \to 1^-}f(x)g(x) &= \lim_{x \to 1^+}f(x)g(x) \\
   e^{-2}(1-a+b) &= 2(1-a+b) \\
   (e^{-2}-2)(1-a+b) &= 0 \\
   (1-a+b) &= 0 \\
   b &= a-1 \\
\hline
   g(x) &= x^2 + ax +(a-1)
\end{align}
