Question

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

Answer #1

Let f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

show that if f is a bounded increasing continuous function on
(a,b), then f is uniformly continuous. Hint: Extend the function to
[a,b].

Let f: (0,1) -> R be uniformly continuous and let Xn be in
(0,1) be such that Xn-> 1 as n -> infinity. Prove that the
sequence f(Xn) converges

Let f : R → R be a continuous function which is periodic. Show
that f is bounded and has at least one fixed point.

Let f : [a, b] → R be bounded, and assume that f is continuous
on [a, b). Prove that f is integrable on [a, b].

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...

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

Let 0 < a < b < ∞. Let f : [a, ∞) → R continuous R at
[a, b] and f decreasing on [b, ∞). Prove that f is bounded
above.

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

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