Question

let F : R to R be a continuous function a) prove that the set {x...

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

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1


0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove or provide a counterexample Let f:R→R be a function. If f is T_U−T_C continuous, then...
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)...
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...
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...
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...
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 A be open and nonempty and f : A → R. Prove that f is...
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.
Let (X, A) be a measurable space and f : X → R a function. (a)...
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.
Prove the following theorem: Theorem. Let a ∈ R and let f be a function defined...
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...
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 −...
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.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT