Question

Consider a closed nowhere dense set E ⊂ [0, 1]. consider a function g : [0, 1] → R prove that if x is not an element of E, then g is continuous at x

Answer #1

Prove:
A nonempty subset C⊆R is closed if and only if
there is a continuous function g:R→R such that
C=g-1(0).

1. A function + : S × S → S for a set S is said to provide an
associative binary operation on S if r + (s + t) = (r + s) +t for
all r, s, t ∈ S. Show that any associative binary operation + on a
set S can have at most one “unit” element, i.e. an element u ∈ S
such that (*) s + u = s = u + s for all...

Let E = {x + iy : x = 0, or x > 0, y = sin(1/x)}. Prove that
the set E is connected, but that is not path-connected or connected
by trajectories
and consider B = {z ∈ C : |z| = 1}. prove ( is easy to see it
but how to prove it ) the following questions ¿is B open?¿is B
closed?¿connected?¿compact?

A function f on a measurable subset E of
Rd is measurable if for all a
in R, the set
f-1([-∞,a)) = {x in
E: f(x) < a} is
measurable
Prove that if f is continuous on
Rd then f is measurable

We say that a set A is dense in a metric space (M,d) provided A
bar = M. Show that if two continuous functions f, g :(M,d) tends to
R on a dense set A, then they must agree on M

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

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 g from R to R is a
differentiable function, g(0)=1, g’(x)>=g(x) for all x>0 and
g’(x)=<g(x) for all x<0. Proof that g(x)>=exp(x) for all x
belong to R.

Consider the function ?:[0,1] → ℝ defined by ?(?) = 0 if ? ∈
[0,1] ∖ ℚ and ?(?) = 1/? if ? = ?/? in lowest terms
1. Prove that ? is discontinuous at every ? ∈ ℚ ∩ [0,1].
2. Prove that ? is continuous at every ? ∈ [0,1] ∖ ℚ

Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...

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