Question

Suppose `I` is an interval and function `f:I->R` and `x in I` . Is it true that `f(x)=1/x` is not bounded function for `I=(0,1)` ?. How do we prove that ?

Answer 1

See explanation...

Explanation:

Given:

`f(x) = 1/x`

`I = (0, 1)`

Then for any `N > 0` we have:

`1/(N+1) in (0, 1)`

`f(1/(N+1)) = N+1 > N`

So `f(x)` is unbounded on `(0, 1)`

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Comments

This example shows the necessity of the condition that the interval be closed in the boundedness theorem:

A continuous function defined on a closed interval is bounded in that interval.