Question

Show that f: R -> R, f(x) = x^2 + 2x is not uniformly continuous on R.

Answer #1

Show that the function f(x) = x^2 + 2 is uniformly continuous on
the interval [-1, 3].

Show that if f and g are uniformly continuous on some interval I
then cf (for all c ∈ R) and f − g are all uniformly continuous on
I

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...

Use
an epsilon-delta argument to show f(x) = 2x^2 - 1 is continuous at
every a ∈ ℝ.

Let f:Ω-->R^m be uniformly continuous on Ω⊂R^n. Show if (Ω)
is bounded, then f(Ω) is bounded.

Show that the function f(x)=x2sin(x) is uniformly
continuous on [0,b] for any constant b>0, but that is not
uniformly continuous on [0,infinity)

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.

A function f : A −→ R is uniformly continuous and its domain A ⊂
R is bounded. Prove that f is a bounded function. Can this
conclusion hold if we replace the "uniform continuity" by just
"continuity"?

show 1/(x^1/2) is not uniformly continuous on the interval
(0,1).

prove that these functions are uniformly continuous on
(0,1):
1. f(x)=sinx/x
2. f(x)=x^2logx

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