# Proving Continuous Functions Cannot Be Two-to-One

• jeff1evesque
In summary, the conversation discusses the proof that a two-to-one function cannot be continuous. The definition of a continuous function is also provided. The attempt at a solution involves using the extreme and intermediate value theorems to show that there exists a y in the range of f with more than two values in its pre-image. However, it is suggested to instead prove that there exist y with only one value in its pre-image.
jeff1evesque

## Homework Statement

Suppose $$f: [0,1] \rightarrow R$$ is two-to-one. That is, for each $$y \in R, f^{-1}({y})$$ is empty or contains exactly two points. Prove that no such function can be continuous.

## Homework Equations

Definition of a continuous function:
Suppose $$E \subset R$$ and f: E \rightarrow R[/tex]. If $$x_0 \in E$$ then f is continuous at $$x_0$$ iff for each $$\epsilon > 0$$, there is $$\varphi >0$$ such that if
$$|x - x_0| < \varphi, x \in E$$,​
then
$$|f(x) - f(x_0)| < \epsilon$$.​

If f is continuous at x for every $$x \in E$$, then we say f is continuous.

## The Attempt at a Solution

I think from an algebraic view, there is more elements in the domain then codomain which means that the function f is not one-to-one, but onto. Though I know this fact, I am pretty sure this will not aid in my attempt to prove this problem. Could someone help me understand why f cannot be continuous.

Thanks,

Jeffrey Levesque

Last edited:
Try assuming that f is continuous and then using the extreme and intermediate value theorems to produce a y in R with more than two values in its pre-image.

Prove, rather, that there exist y such that $$\displaystyle f^{-1}(\left{y\right})$$ contains exactly one value.

## What does it mean for a function to be "continuous"?

A continuous function is a mathematical function where the output value changes gradually as the input value changes. This means that small changes in the input result in small changes in the output, and there are no sudden jumps or gaps in the graph of the function.

## What is a "two-to-one" function?

A two-to-one function is a mathematical function where every output value is associated with exactly two input values. In other words, for every x-value, there are exactly two y-values that satisfy the function.

## Why is it impossible for a continuous function to be two-to-one?

This is because a continuous function must follow the property of the intermediate value theorem, which states that if a function is continuous on a closed interval, it must take on every value between its minimum and maximum values. If a function is two-to-one, then it would not be able to satisfy this property, as there would be at least one output value that is not associated with any input value.

## Can you give an example of a continuous function that is not two-to-one?

One example is the function f(x) = x^2, which is continuous on the entire real number line. However, for every positive output value, there are two input values (one positive and one negative) that satisfy the function, so it is not two-to-one.

## Are there any exceptions to this rule?

No, there are no exceptions to this rule. All continuous functions must follow the intermediate value theorem, and therefore cannot be two-to-one.

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