Question

Let f and g be continuous functions from C to C and let D be a dense

subset of C, i.e., the closure of D equals to C. Prove that if f(z) = g(z) for

all x element of D, then f = g on C.

Answer #1

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Let f, g : X −→ C denote continuous functions from the open
subset X of C. Use the properties of limits given in section 16 to
verify the following:
(a) The sum f+g is a continuous function. (b) The product fg is
a continuous function.
(c) The quotient f/g is a continuous function, provided g(z) !=
0 holds for all z ∈ X.

Let D ⊆ R, a ∈ D, let f, g : D −→ R be continuous functions. If
limx→a f(x) = f(a) and limx→a g(x) = g(a) with f(a) < g(a), then
there exists δ > 0 such that x ∈ D, 0 < |x − a| < δ =⇒
f(x) < g(x).

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

Let C [0,1] be the set of all continuous functions from [0,1] to
R. For any f,g ∈ C[0,1] define dsup(f,g) =
maxxE[0,1] |f(x)−g(x)| and d1(f,g)
= ∫10 |f(x)−g(x)| dx. a) Prove that for any
n≥1, one can find n points in C[0,1] such that, in dsup
metric, the distance between any two points is equal to 1. b) Can
one find 100 points in C[0,1] such that, in d1 metric,
the distance between any two points is equal to...

Let f and g be continuous functions on the reals and let S={x in
R | f(x)>=g(x)} . Show that S is a closed set.

Let f(z) and g(z) be entire functions, with |f(z) - g(z)| < M
for some positive real number M and all z in C. Prove that f'(z) =
g'(z) for all z in C.

Let f and g be measurable unsigned functions on R^d . Assume
f(x) ≤ g(x) for almost every x. Prove that the integral of f dx ≤
Integral of g dx.

1. Let A = {1,2,3,4} and let F be the set of all functions f
from A to A. Prove or disprove each of the following
statements.
(a)For all functions f, g, h∈F, if f◦g=f◦h then g=h.
(b)For all functions f, g, h∈F, iff◦g=f◦h and f is one-to-one
then g=h.
(c) For all functions f, g, h ∈ F , if g ◦ f = h ◦ f then g =
h.
(d) For all functions f, g, h ∈...

Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.

Let f and g be functions between A and B. Prove that f = g iff
the domain of f = the domain of g and for every x in the domain of
f, f(x) = g(x).
Thank you!

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