Question

$f: \Bbb R \to \Bbb R $ is continuous. $ A = \{ T > 0 : f(x+T) = f(x) , \forall x \in \Bbb R \}$ not empty. Prove $f$ is constant $\iff$ $inf A=0$.

score 54 · Answer 1 · 2021-06-25 13:43:24

$f: \Bbb R \to \Bbb R $ is continuous. $ A = \{ T > 0 : f(x+T) = f(x) , \forall x \in \Bbb R \}$ not empty. Prove $f$ is constant $\iff$ $inf A=0$.

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List Question

$f: \Bbb R \to \Bbb R $ is continuous. $ A = \{ T > 0 : f(x+T) = f(x) , \forall x \in \Bbb R \}$ not empty. Prove $f$ is constant $\iff$ $inf A=0$.

Is the supremum of a sequence of random variables a random variable?

How does Holder's inequality apply to the expectation operator when using the infinity norm?

Can the supremum and (measure theoretic) integral be interchanged?

Supremum of a set equals to infinity

Proving the least upper bound of a subset X of 2^S

A question on completeness condition from Ahlfors' book

Existence of an Improper Integral which is bounded.

Confusion about alternative unit interval connectedness proof

Verification of proof that $\sup(A) = - \inf(-A)$

Value of $I=\inf_{f\in S} \int_0^1 f(x)^2 \, \text{d}x$

Proving convergence of a sequence of suprema in exercise 2.4.7 of Understanding Analysis by Abbott (2015).

The infimum of a semi-continuous function on a compact set

Likelihood ratio hypothesis test for the exponential distribution.

Validation of Proof that Nested Interval Property implies Axiom of Completeness

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