Question

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

Answer #1

Prove the conjecture or provide a counterexample:
Let U ∈ in the usual topology, and let F be a finite set. Then
(U−F) ∈ in the usual topology.

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 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.

Let f be a continuous function on the real line. Suppose f is
uniformly continuous on the set of all rationals. Prove that f is
uniformly continuous on the real line.

Let A be open and nonempty and f : A → R. Prove that f is
continuous at a if and only if f is both upper and lower
semicontinuous at a.

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.

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

. Let f and g : [0, 1] → R be continuous, and assume f(x) = g(x)
for all x < 1. Does this imply that f(1) = g(1)? Provide a proof
or a counterexample.

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

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

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