Question

A function f : R → R is called additive if f(x + y) = f(x) + f(y) for all x, y ∈ R. Is the set of all additive functions a subspace of F(R, R)? Give a proof of counter example.

Answer #1

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

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

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.

1)T F: All (x, y, z) ∈ R 3 with x = y + z is a subspace of R 3
9
2) T F: All (x, y, z) ∈ R 3 with x + z = 2018 is a subspace of R
3
3) T F: All 2 × 2 symmetric matrices is a subspace of M22. (Here
M22 is the vector space of all 2 × 2 matrices.)
4) T F: All polynomials of degree exactly 3 is...

(1) (5 pts) Consider the function f : + ×+ → given by f(x, y) =
x! y!(x−y)! . Where and x and y are positive integers h Hint: this
is the combination formula, x y i (a) What types of relationships
are generated by this function, please justify your answers using
examples or counter examples. (b) How many combinations of 2 pairs
can be generated from a power of R, assuming there are 4 element in
set R .

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

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.

Evaluate the double integral for the function
f(x,
y)
and the given region R.
f(x, y) =
5y + 5x;
R is the rectangle defined by
5 ≤ x ≤ 6
and
2 ≤ y ≤ 4

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