Strange proposition in calculus

In summary, the problem is asking to prove that the range of an injective and continuous function on the interval [a,b] is equal to the interval between the function values at the endpoints, f(a) and f(b). This can be done using the Intermediate Value Theorem. The initial attempt at a solution was incorrect, but the correct interpretation of the question is to show that the range on [a,b] is indeed [f(a),f(b)].
  • #1
Speedholic
3
0

Homework Statement



f(x) is an injective function (1 to 1) and continuous in [a, b], and f(a) < f(b). Show that the
range of f is the interval [f(a), f(b)]

Homework Equations



Intermediate Value Theorem

The Attempt at a Solution


We are asked to use the intermediate value theorem to prove it. However, it seems to me that the proposition is false.

Suppose f(x) = x, a = 0, b = 1. f(x) is 1 to 1, continuous in [a,b] and f(a) < f(b), but its range is (-inf, inf), not [0, 1].

Am I reading this question wrong??
 
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  • #2
I think the question is asking you to show that the range on [a,b] is [f(a),f(b)]
 
  • #3
Ah... that makes sense...
 

FAQ: Strange proposition in calculus

What is a "strange proposition" in calculus?

A "strange proposition" in calculus refers to a statement or theorem that may seem counterintuitive or unusual at first glance, but can be proven to be true using mathematical principles and logic.

Can you provide an example of a strange proposition in calculus?

One example of a strange proposition in calculus is the Banach-Tarski paradox, which states that a solid sphere can be divided into a finite number of pieces and rearranged to form two identical copies of the original sphere. This goes against our intuitive understanding of volume and seems impossible, but it has been proven using mathematical concepts such as non-measurable sets.

Why are strange propositions important in calculus?

Strange propositions challenge our understanding of mathematics and push us to think critically and creatively. They also help to expand our knowledge and discover new concepts and principles that can be applied in various fields of science and technology.

What is the process for proving a strange proposition in calculus?

The process for proving a strange proposition in calculus involves using mathematical reasoning and techniques, such as induction, contradiction, or direct proof. It also requires a deep understanding of the underlying concepts and principles involved.

Are there any real-world applications of strange propositions in calculus?

Yes, there are many real-world applications of strange propositions in calculus, particularly in fields such as physics, engineering, and computer science. For example, the Banach-Tarski paradox has implications for the study of non-Euclidean geometry and has been used to develop algorithms for computer graphics and animation.

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