Prove that f must be a constant function

In summary, the conversation discusses how to prove that a continuous function with a range contained in the set of integers must be a constant function. The use of proof by contradiction and epsilon-delta proof is mentioned, and the possibility of using the intermediate value theorem is also brought up. Eventually, it is concluded that the proof is possible by considering the difference between two non-equal values of the function.
  • #1
SithsNGiggles
186
0

Homework Statement



If f: ℝ → ℝ is a continuous function with the property that its range is contained in the set of integers, prove that f must be a constant function.

Homework Equations



The Attempt at a Solution



I know why this is true, I just don't know how to begin an actual proof. So far I've thought of proving by contradiction, with letting f be discontinuous and use f(x) = [x], whose range set is contained in Z.

I seem to have trouble with the format of a formal proof.
 
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  • #2
OK, suppose f(x) and f(y) are not equal for some x and y. Now, you know that their difference must be at least 1, right? So, let [itex]\epsilon = 1[/itex] and...

Do you see how this might work?
 
  • #3
It seems an epsilon-delta proof might work well here.
 
  • #4
Are you allowed to use intermediate value theorem?
 
  • #5
Oh, ok I think I got it. Thanks.
 

FAQ: Prove that f must be a constant function

What does it mean for a function to be constant?

A constant function is a mathematical function that always produces the same output value for a given input value. In other words, the value of the function does not change as the input value varies.

How can you prove that a function is constant?

To prove that a function is constant, you must show that the output value is the same for all possible input values. This can be done by evaluating the function at multiple input values and showing that the output is always the same.

What is the significance of proving that f is a constant function?

Proving that f is a constant function can be useful in a variety of mathematical and scientific contexts. It can help simplify calculations, identify patterns, and provide a deeper understanding of the behavior of a function.

Can a non-linear function be constant?

No, a non-linear function cannot be constant. A constant function is always a straight horizontal line, whereas a non-linear function has a curved or non-uniform graph.

Are there any exceptions to the rule that a constant function must have a horizontal line graph?

Yes, there are a few exceptions. A constant function can also have a vertical line graph or a single point graph, depending on the definition of the function and its domain and range.

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