Can a Continuous Function Map One Value to Two Different Points?

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Homework Help Overview

The discussion revolves around the properties of continuous functions, specifically addressing the assertion that a continuous function cannot map a single value to two different points. The original poster seeks to demonstrate this property for all real numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the use of contradiction and the intermediate value theorem as potential approaches. There is also a discussion about the interpretation of the problem, particularly regarding the meaning of "two solutions" in the context of continuous functions.

Discussion Status

The conversation is ongoing, with participants questioning the clarity of the original statement and offering insights into the nature of the function being discussed. Some guidance has been provided regarding the use of contradiction and the implications of mapping in the context of continuous functions.

Contextual Notes

There appears to be some ambiguity in the original poster's phrasing, leading to questions about the precise nature of the proof being sought. Participants are clarifying the definitions and assumptions related to the problem.

za10
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Homework Statement



Show that a continuous function such that for all c in the reals, the equation f(x) = c cannot have two solutions

Homework Equations





The Attempt at a Solution



I was thinking along the lines of a contradiction or somehow using intermediate value theorem but it seems like it is so easy it is hard.
 
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What are you actually trying to prove here? Your sentence doesn't parse properly
 
show that a continuous function cannot have two solutions for the equation f(x) = c for every c.
 
I have f(x) = c... am I solving for x given c? What you're trying to say is that f can't be two to one (i.e. for every point p in the image, there are two points in the preimage of p).

Contradiction is a good place to start. There have to be two points that f maps to zero, consider f on the interval between them
 

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