For what values of a and b that make the function f continuous everywhere.

Question

f(x)={(x^2-4)/(x-2) if x<2
ax^2-bx-18 if 2<x<3
10x-a+b if x>3

Answer 1

There are three segments of functions, and two possibly discontinuous points, at x=2 and x=3.

Equate the adjacent segments to find the value of a and b that make the segments have the same values at x=2 and x=3.

Finally, check that the limits approaching from each side (2- and 2+, 3- and 3+) of the combined function are identical. They should be if the function is continuous.