Question

if f: D - R be continuous, and D is close, then F(D) is closed. prove or give counterexample

Answer #1

the statement is false

so for counter example take a sequence in f(D) and start the proof

Prove or give a counterexample: If f is continuous on R and
differentiable on R∖{0} with limx→0 f′(x) = L, then f is
differentiable on R.

Prove or provide a counterexample
Let f:R→R be a function. If f is T_U−T_C continuous, then f is
T_C−T_U continuous.
T_U is the usual topology and T_C is the open half-line
topology

Prove or give a counter example: If f is continuous on R and
differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f
is differentiable on 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

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).

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...

2. Suppose [a, b] is a closed bounded interval. If f : [a, b] →
R is a continuous function, then prove f has an absolute minimum on
[a, b].

Prove the IVT theorem
Prove: If f is continuous on [a,b] and f(a),f(b) have different
signs then there is an r ∈ (a,b) such that f(r) = 0.
Using the claims:
f is continuous on [a,b]
there exists a left sequence (a_n) that is increasing and
bounded and converges to r, and left decreasing sequence and
bounded (b_n)=r.
limf(a_n)= r= limf(b_n), and f(r)=0.

Use each definition of a continuous function to prove that every
function f: Z --> R is
continuous

Let D ⊆ R, a ∈ D, let f, g : D −→ R be continuous functions. If
limx→a f(x) = f(a) and limx→a g(x) = g(a) with f(a) < g(a), then
there exists δ > 0 such that x ∈ D, 0 < |x − a| < δ =⇒
f(x) < g(x).

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