Answer to Question #16429 in Functional Analysis for Ciaran Duffy

Question

Suppose the function f is continuous on the interval [a, b] and never zero on [a, b].
Is it possible that f(z) < 0 for some z ∈ [a, b] and f(w) > 0 for some w ∈ [a, b]?
Explain your answer.

Answer 1

This is not impossible due the intermediate value theorem.
This theorem
claims that if a continuous function
f : [z,w] --> R
on the interval
[z,w] is such that
f(z) and f(w) are non-zero and have distinct signs,
e.g.
either
f(z)>0 and f(w)<0
or
f(z)<0 and
f(w)>0,
then there exists a point c in [z,w] such that f(c)=0.

In
our case suppose
f(z)<0 for some z ∈ [a, b] and f(w)>0 for some w ∈
[a, b]
We can assume that z<w.
Then by the above theorem ther eshould
exists a point c in [z,w] such that f(c)=0.
But c also belongs to [a,b],
and so f is not never zero on [a,b].
This gives a contradiction, and so

either
f>0
or
f<0
of all of [a,b].