
“real: chapter_09” — 2011/5/22 — 23:21 — page 38 — #38
9-38 Real Analysis
UNSOLVED EXERCISES
1. Let f , f
1
, f
2
, ... be real-valued functions defined on a compact metric
space (X , d) such that x
n
→ x in X implies f
n
(x
n
) → f (x) as n →∞
in R.Iff is continuous, then show that the sequence of functions {f
n
}
converges uniformly to f .
2. For a sequence {f
n
} of real-valued functions defined on a topological
space X that converges uniformly to a real-valued function f on X ,
establish the following:
(a) If x
n
→ x as x →∞and f is continuous at x, then f
n
(x
n
) → f (x)
as n →∞.
(b) If each f
n
is continuous at some point x
0
∈ X , then f is also
continuous at the point ...