Just had this problem on a test and totally blanked out.

x² - 4
x - 2           if x ≠ 2

m             if x ≠ 2

For what value of m is the function continuous on (-∞ , ∞)?

Notice that (x² - 4) / (x - 2) = x + 2 for any x ≠ 2.

Hence, to be continuous at x = 2, m must be equal to 4.

I think you missed some information, it would make more sense if f(x) = (x^2-4)/(x-2) if x≠2 or m if x=2 in which case you compute the limit of the function for x -> 2 by the left and x -> 2 by the right which is 4 because
(x^2-4)/(x-2) => (x+2)(x-2)/(x-2) => x+2 so the limit in 2 is 4.

Thus the function is continuous for m=4

Oh, really? That's all huh? I think I actually got the right answer then.

Also m0hawk you are correct, the second one was supposed to say x=2.