Let Find a+b if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper).

Put 2 into  each of the  first two functions  for x and set them  equal

a(2) + 3 = 2 - 5

2a = 2 - 5 - 3

2a = -6

a = -3

Put - 2  into  each of the last two  functions for x and  set them equal

-2 - 5 = 2(-2) - b

-7 = -4 - b

b = -4 + 7

b = 3

a + b =  3 + -3  =   0

Rigorously, continuity is defined below:
 

For every $k$ in the domain of $f$;

• $f(k)$ exists
•  $\lim_{i\to k}f(i)$ exists (i.e. $\lim_{i\to k^-}f(i) = \lim_{i \to k+}f(i)$)
• and $\lim_{i\to k}f(i) = f(k)$