Can this function still be a constant function?

In summary, the discussion revolves around whether a given mathematical function can maintain the characteristics of a constant function under various conditions or transformations. It explores the criteria that define constant functions and examines potential scenarios where a function might appear constant but may not meet the strict definition. The analysis highlights the importance of understanding function properties in different contexts.
  • #1
tellmesomething
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Homework Statement
Let for all where is continuous function & if then may be?
Relevant Equations
None
So I know that since that means can achieve all possible values on the real number line meaning is a constant function. And I know hwo to calculate the limit beyond that. However my teacher made a point which I dont necessarily agree with he said, if wasn't continuous we could not have said Its a constant function.
He gave an example like
for all
for all
The above piecewise function also satisfies

But my doubt is since the original question already says that , isnt that enough information to conclude its a constant function.
What I mean is if the question was instead
Let for all & , if then may be?
Could we still not have concluded that is a constant function I.e on a graph it would be parallel to the axis for alll ?

Because for all I know, a continuous function means that the limiting value and the functional value are equal , and here since we already know that the function for all we know that the limiting value and the functional value are equal..
 
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  • #2
I do not understand why you come up with such a complicated example or what the roles of or are. I don't even understand what is, or how the function in the limit expression is defined.

tellmesomething said:
But my doubt is since the original question already says that x€R , isnt that enough information to conclude its a constant function.
Obviously not. What's wrong with your teacher's example? It is not constant, not continuous, its domain is its codomain if and if This example uses the fact that
 
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  • #3
fresh_42 said:
I do not understand why you come up with such a complicated example or what the roles of or are. I don't even understand what is,
Sorry I didnt realise some variables got capitalised. I have edited it now.
fresh_42 said:
or how the function in the limit expression is defined.
I dont understand this though, if you could clear it out.
fresh_42 said:
Obviously not. What's wrong with your teacher's example? It is not constant, not continuous, its domain is its codomain if and if This example uses the fact that

I think I get it now. Please check .. Firstly if then I believe as well I.e all real numbers are in the set of as well as in the set of
You obviously cannot get an irrational number when you multiply a rational number by 2 or vice versa. Domain of the function is all real numbers.
implies if we get a constant function and if we get a different constant function. And if we want these constant functions to be equal we need to declare that its a continuous function overall.
 
  • #4
tellmesomething said:
I dont understand this though, if you could clear it out.
I meant this monster here:

You haven't said what is. If it was meant to be then why write ?

tellmesomething said:
I think I get it now. Please check .. Firstly if then I believe as well I.e all real numbers are in the set of as well as in the set of
You obviously cannot get an irrational number when you multiply a rational number by 2 or vice versa. Domain of the function is all real numbers.
implies if we get a constant function and if we get a different constant function. And if we want these constant functions to be equal we need to declare that its a continuous function overall.
Yes. Continuity guarantees that if i.e. the function values and cannot "jump" if and are close enough. We can always find a rational and an irrational number that are arbitrarily close, so their function values have to be, too.

The standard proof of such a statement would be to prove that is constant on then that it is constant on say and continuity allows us to conclude:

Let and a sequence of rational numbers such that Since is dense, such a sequence exists. Continuity of now says
 
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  • #5
fresh_42 said:
I meant this monster here:

You haven't said what is. If it was meant to be then why write ?
This is how the question was framed. From what I gather if it said directly it would not be "tricky" as the whole point is to conclude from the question that is a constant and its value will be same as which is . Beyond that its pretty simple to calculate the limit which would have been the same if x was approaching something like π.

This might seem like a very silly question understandably so its for high school calc.

fresh_42 said:
Yes. Continuity guarantees that if i.e. the function values and cannot "jump" if and are close enough. We can always find a rational and an irrational number that are arbitrarily close, so their function values have to be, too.

The standard proof of such a statement would be to prove that is constant on then that it is constant on say and continuity allows us to conclude:

Let and a sequence of rational numbers such that Since is dense, such a sequence exists. Continuity of now says
Though I dont understand a bit of the notation, I think I get the gist of it. Many thanks!
 
  • #6
tellmesomething said:
This might seem like a very silly question understandably so its for high school calc.
Continuity and limits at the high school level? That's ambitious, especially if all this should be rigorous. I would like to see the proof for
 
  • #7
fresh_42 said:
Continuity and limits at the high school level? That's ambitious, especially if all this should be rigorous. I would like to see the proof for
I am not at all familiar with proofs to do this any justice. But from the discussion we had,
Since is defined for all
We know that is also defined for all as
If we suppose x is any number y, we know that it can be written as and multiplying it with 2 would return us the number hence telling us that all numbers in the set of x are also in the set of 2x

Now its given that which means the subset of all rational numbers' functional value is constant and the subset of all irrational numbers' functional value is constant. Since the function is continuous I.e has no breaks in the graph this would mean that very close to any number on the real number line the functional value has to be equal to the functional value at the number. And we know that there are infinite number of irrational as well as rational values near any number we can conclude that the function is constant overall.

This would mean that = =

I think I just reiterated everything I gained from your posts. I know you asked of me a formal proof but we dont do proof writing until college here so this is the only hand wavy thing I know..
 
  • #8
To show the function is constant, you could note that
And use the continuity of to show that .
 
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  • #9
Are you sure it isn't ? That would make sense.

Let's set Then On the other end we get Since the sequence get's closer and closer to we have by continuity of (no jumps). Now we can set into the formula for
 
  • #10
fresh_42 said:
Are you sure it isn't ? That would make sense.

Let's set Then On the other end we get Since the sequence get's closer and closer to we have by continuity of (no jumps). Now we can set into the formula for
I think I didnt follow. I thought a lot about this but I dont know..
Isn't f a constant function? How does it matter if its 2024 or 2048.. The question says 2024 by the way..
 
  • #11
tellmesomething said:
I think I didnt follow. I thought a lot about this but I dont know..
Isn't f a constant function? How does it matter if its 2024 or 2048.. The question says 2024 by the way..
I still don't know whether the condition is sufficient to prove that is constant. I haven't seen a proof. If it is constant, then it doesn't matter since However, I do not know.

If is not constant, but and then this would be sufficient to prove We get from that for all integers This means that in case we would have such a condition. Going on with that argument, we also get

and by continuity of we get

and

Conclusion: I can prove that if without knowing that is constant. If I only have then I do not know that because I haven't seen a proof that is constant.
 
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  • #12
fresh_42 said:
I still don't know whether the condition is sufficient to prove that is constant. I haven't seen a proof.
Continuity at is sufficient:

PeroK said:
To show the function is constant, you could note that
And use the continuity of [edit: at 0] to show that .
 
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FAQ: Can this function still be a constant function?

1. What is a constant function?

A constant function is a type of function that always produces the same output value, regardless of the input value. Mathematically, it can be expressed as f(x) = c, where c is a constant real number.

2. Can a function be constant if it has variable inputs?

Yes, a function can still be considered a constant function even if it has variable inputs, as long as the output remains the same for all input values. For example, f(x) = 5 is a constant function regardless of the value of x.

3. How can I determine if a function is constant?

You can determine if a function is constant by evaluating it at different input values. If the output does not change for any of the inputs, then the function is constant. Additionally, the graph of a constant function is a horizontal line.

4. Are there any exceptions to what makes a function constant?

Yes, a function that has different outputs for different inputs is not a constant function. Additionally, if a function is defined piecewise and one of the pieces is a constant, the overall function may not be constant if the other pieces vary.

5. Can a constant function be represented in different forms?

Yes, a constant function can be represented in various forms, such as an equation (e.g., f(x) = 7), a graph (horizontal line at y = 7), or a table of values (showing the same output for different inputs). However, regardless of the representation, it remains a constant function as long as the output is unchanged.

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