Proving a function to be constant

In summary, the question states that suppose f is continuous and that f(x) in the set Q (f(x)eQ) for all x in the Reals (xeR). Prove that f is constant.
  • #1
Halen
13
0
The question states:
Suppose that f:R--->R is continuous and that f(x) in the set Q (f(x)eQ) for all x in the Reals (xeR)
Prove that f is constant.

How would you go about this question? Any help is appreciated!

i know we have to prove that f(x) is equal to some constant (p/q) ; q not equal to 0 for all xeR but how would you prove this?
 
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  • #2
Do you know anything about topology? If so, what do you know about R? And what do you know about what f(R)? What does f(R) have that Q doesn't?
 
  • #3
no, unforunately, i don't do topology..

but i can try answering the question..

Has it anything to do with f(Q) being countable?
 
  • #4
I never said anything about f(Q). I said about f(R). But if you don't know some basic topology, then it'd be harder to explain. I think we can side-step it if with a certain equivalent.

Since f is continuous, it satisfies the intermediate value theorem. Why is this a problem?
 
  • #6
thank you! that is great help indeed!

i have started the proof.. am i going on the right track?

suppose there exists a,b e R such that f(a), f(b) e Q.
WLOG, f(a)<f(b)
Between any two rational numbers, there is an irrational number, say e such that f(a)<e<f(b)
by IVT, there exists ce(a,b) such that f(c)=e

now the remaining part is to prove f(c)=e.. is that right?
 
  • #7
Halen said:
now the remaining part is to prove f(c)=e.. is that right?

You've already proved that f(c) = e from the IVT. You're trying to reach a contradiction since this is a proof by contradiction. What's the contradiction from f(c) = e?
 
  • #8
ohh.. the contradiction would be that f(c) is rational but e is not..
 
  • #9
Halen said:
ohh.. the contradiction would be that f(c) is rational but e is not..

no

f(c) = e is irrational. What do we know about the function f from the hypothesis?
 
  • #10
the function f is continuous and has a domain and range in the reals.
 
  • #11
Halen said:
the function f is continuous and has a domain and range in the reals.

Read it again. You typed it in the first post! The domain is the real numbers, but what is the range?
 
  • #12
i really hope I'm right now.. :) is it f(x)eQ?
 
  • #13
Halen said:
i really hope I'm right now.. :) is it f(x)eQ?

Yes, so what's the contradiction here?
 
  • #14
so there does not exist a point such that f(c)=e because it is irrational but we are told f(x)eQ? and so f is constant
 
  • #15
Halen said:
so there does not exist a point such that f(c)=e because it is irrational but we are told f(x)eQ? and so f is constant

You got it. :approve:
 
  • #16
thank you! finally! :)
 

What does it mean to prove a function to be constant?

Proving a function to be constant means showing that the output of the function is the same for all input values. In other words, the function does not change or vary with different inputs.

How do you prove a function to be constant?

The most common method for proving a function to be constant is by using the definition of a constant function, which states that a function f(x) is constant if and only if f(x) = k for all values of x, where k is a constant. To prove this, you must show that the output of the function is equal to a constant value for all inputs.

What is an example of a constant function?

An example of a constant function is f(x) = 5. This function will always output the value 5, regardless of the input x. Another example is the identity function f(x) = x, which has an output equal to its input for all values of x.

Can all functions be proven to be constant?

No, not all functions can be proven to be constant. Some functions, such as exponential functions or trigonometric functions, have varying outputs for different inputs and cannot be considered constant.

Why is proving a function to be constant important?

Proving a function to be constant is important in mathematics and science because it allows us to identify and analyze relationships between variables. It also helps to simplify calculations and make predictions based on the behavior of the function.

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