Question

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).

Answer #1

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...

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 : 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 defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

Let f : R − {−1} →R be defined by f(x)=2x/(x+1).
(a)Prove that f is injective.
(b)Show that f is not surjective.

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

prove that this function is uniformly continuous on
(0,1):
f(x) = (x^3 - 1) / (x - 1)

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.

Is the function f : R → R defined by f(x) = x 3 − x injective,
surjective, bijective or none of these?
Thank you!

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