Proving Uniqueness in Continuous Functions with Positive Values

[proximaankit]

Homework Statement

Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely
any such τ corresponds to a unique t. Provide a brief explanation on why there is such a 1-1 correspondence.

Homework Equations

NA

The Attempt at a Solution

Stuck on it but here are some of my thoughts and reasoning:

I first view τ as function dependent upon t. since k(t) is positive and continuous, that will mean that the antiderivative of k(t) will only give us increasing values for increasing t. The new k(η) function is essentially same as k(t) except with η as the independent var. Hence since the k(t) is positive then k(η) is also positive. Then the integral of k(η) must be increasing for each increasing t. Hence for t₂ and t₁ the integral of k(η) from 0 to t₂ is greater than the integral of k(η) from 0 to t₁. This makes sure the for every different t substitute into the integral have a different output. And as we said τ is


The problem is how do I show the unique τ for each t part.

Thank you very much in advance for any help :)

 

[STEMucator]

Homework Statement

Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely
any such τ corresponds to a unique t. Provide a brief explanation on why there is such a 1-1 correspondence.

Homework Equations

NA

The Attempt at a Solution

Stuck on it but here are some of my thoughts and reasoning:

I first view τ as function dependent upon t. since k(t) is positive and continuous, that will mean that the antiderivative of k(t) will only give us increasing values for increasing t. The new k(η) function is essentially same as k(t) except with η as the independent var. Hence since the k(t) is positive then k(η) is also positive. Then the integral of k(η) must be increasing for each increasing t. Hence for t₂ and t₁ the integral of k(η) from 0 to t₂ is greater than the integral of k(η) from 0 to t₁. This makes sure the for every different t substitute into the integral have a different output. And as we said τ is


The problem is how do I show the unique τ for each t part.

Thank you very much in advance for any help :)

Is this supposed to be an if and only if or are you to prove that for any $t$ there is a unique $τ$?

If it's the latter, you should start by assuming there are two unique $τ$ and then show that they both must be the same.