Question

let F : R to R be a continuous function

a) prove that the set {x in R:, f(x)>4} is open

b) prove the set {f(x), 1<x<=5} is connected

c) give an example of a function F that {x in r, f(x)>4} is disconnected

Answer #1

Prove or provide a counterexample
Let f:R→R be a function. If f is T_U−T_C continuous, then f is
T_C−T_U continuous.
T_U is the usual topology and T_C is the open half-line
topology

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

Let X be a set and A a σ-algebra of subsets of X.
(a) A function f : X → R is measurable if the set {x ∈ X : f(x)
> λ} belongs to A for every real number λ. Show that this holds
if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ
∈ R. (b) Let f : X → R be a function.
(i) Show that if...

Prove or give a counter example: If f is continuous on R and
differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f
is differentiable on R .

Problem 2. Let F : R
→ R be any function (not necessarily measurable!).
Prove that the set of points x ∈ R such
that
F(y) ≤ F(x) ≤
F(z)
for all y ≤ x and z ≥ x is
Borel set.

Let (X, A) be a measurable space and f : X → R a function.
(a) Show that the functions f 2 and |f| are measurable whenever
f is measurable.
(b) Prove or give a counterexample to the converse statement in
each case.

Let A be open and nonempty and f : A → R. Prove that f is
continuous at a if and only if f is both upper and lower
semicontinuous at a.

Prove the following theorem:
Theorem. Let a ∈ R
and let f be a function defined on an
interval centred at a.
IF f is continuous at a
and f(a) > 0 THEN
f is strictly positive on some interval
centred at a.

Let f be a continuous function on the real line. Suppose f is
uniformly continuous on the set of all rationals. Prove that f is
uniformly continuous on the real line.

Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −
y|^{1/2} for all x, y ∈ R. Apply E − δ definition to show that f is
uniformly continuous in R.

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