Prove f(y) = y for every real number y

In summary, proving f(y) = y means showing that the output of a function will always be equal to the input for any value of y. This is typically done using mathematical techniques and can help us understand the behavior of a function. However, there may be cases where f(y) = y is not true, such as when there are restrictions on the domain or range of the function.
  • #1
Annie B
1
0
A function f: R->R is a continuous function such that f(q) = q for every rational number q.
Prove f(y) = y for every real number y.

I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?
 
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  • #2
I moved your thread to our homework forums.

You can use exactly that property about limits, together with continuity.
 

Related to Prove f(y) = y for every real number y

1. What does it mean to "prove f(y) = y"?

To prove f(y) = y means to show that for every real number y, the output of the function f will always be equal to y. This means that no matter what value of y is plugged into the function, the resulting output will always be the same as the input.

2. How do you prove f(y) = y?

To prove f(y) = y, you would need to use mathematical techniques such as algebra or calculus to manipulate the function and show that its output will always be equal to the input. This may involve solving for y or showing that the function follows certain properties, such as the identity property.

3. Can you provide an example of a function that satisfies f(y) = y?

One example of a function that satisfies f(y) = y is the identity function, where f(y) = y for all real numbers y. This means that no matter what value of y is plugged into the function, the output will always be equal to the input.

4. Why is proving f(y) = y important in mathematics?

Proving f(y) = y is important in mathematics because it allows us to show that a function is consistent and follows certain rules or properties. This can help us understand the behavior of the function and make predictions about its outputs for different inputs.

5. Are there any cases where f(y) = y may not be true?

Yes, there may be cases where f(y) = y is not true. This could happen if the function is not defined for certain values of y or if there are restrictions on the domain of the function. It is important to carefully consider the domain and range of a function when trying to prove f(y) = y for every real number y.

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