Question

Let S1 be the unit circle. Prove that any continuous function f : S1 → S1 has at least a fixed point. Find this point.

Answer #1

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 : [1, 2] → [1, 2] be a continuous function. Prove that
there exists a point c ∈ [1, 2] such that f(c) = c.

Suppose f : [a, b] → [a, b] is a continuous function. Prove that
it has a fixed point x (that is, a point x such that f(x) = x).

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 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.

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

if
the function f is differentiable at a, prove the function f is also
continuous at a.

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...

Let f be a continuous function. Suppose theres a sequence (x_n)
in [0,1] where lim f(x_n))=5. Prove there is a point x in [0,1]
where f(x)=5.

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.

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