
“real: chapter_05” — 2011/5/22 — 22:55 — page 13 — #13
Limits and Continuity 5-13
by (f ± g)(x) = f (x) ± g(x), (fg)(x) = f (x)g(x) are all continuous
at x = a. In addition if g(x) = 0 for x ∈ S then the function f /g
defined by
f
g
(x) =
f (x)
g(x)
is also continuous at x = a.
Proof Follows easily by using Theorem 5.2.13 and the formulation of
continuity using limits.
Theorem 5.3.9 Let f : S → T, g : T → R be functions. Let a ∈ S, f
be continuous at a and g be continuous at f (a). Then g ◦f is continuous
at a.
Proof Let x
n
→ a as n →∞. It is clear that f (x
n
) → f (a) as n →∞
(using continuity of f at a) and g(f (x
n
)) → g(f (a)) as n →∞(using
continuity