A note on trying to extend the intermediate value theorem
First, it is necessary to introduce the following definitions1,
-
The function is said to be increasing at
if for all -values in some interval about it is true that when then , and when then . -
The function is said to be decreasing at
if for all -val ues in some interval about it is true that when then , and when then .
Then the definition of “a function is non-decreasing at
-
The function is said to be non-decreasing at
if for all -values in some interval about it is true that when then , and when then .
The intermediate value theorem states:
If
is a continuous function on a closed interval , and if is any value strictly between and , that is , then for some in .
My original conjecture was to extend it to:
If
is a continuous function on a closed interval with , and if is any value strictly between and , that is , then for some in and is non-decreasing at the .
But it was said that the Weierstrass function, as a counterexample, is indeed nowhere monotonic, while I still couldn’t understand that. After reading the following proof for the intermediate value theorem2,
I realized I could save the conjecture to:
If
is a continuous function on a closed interval with , and if is any value strictly between and , that is , then there is a in ) in cause , and such that
in every left neighborhood of
, there is always a such that . for each
in any right neighborhood of . A similar argument could be drawn for
.
The following two functions, which are both differentiable and continuous in neighborhoods around 0, are helpful for my consideration about the question.
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