Integral Problem: Non-Zero Continuous Function R to R Solution

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In summary, the conversation discusses the existence of a non-zero continuous function, f, from R to R such that the integral of f(x) over the interval (c, c^2+1) is equal to 0 for all values of c in R. The conversation also explores different methods for approaching this problem, including finding a primitive function F(c) and using the Lebesque function. However, it is determined that no such function can exist due to the properties of its derivative.
  • #1
Wicketer
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Is there a non-zero continuous function, f, R -> R such that ∫f(x) dx over (c, c^2+1) = 0, for all c in R? Hope this makes sense. I've been trying to find a more formal way of approaching this rather than just visualizing it with graphs. Any thoughts?

Many thanks.
 
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  • #2
Try to see if you can built a primitive F(c) such that F(c²+1)=F(c) .
If you can find one, then f is its derivative.

I guess it might be useful to observe that c²+1 > c .
Defining F in the interval [0,1] might be all that is necessary.
Will F be continuous, will its derivative be continuous, ?

Did you give the full statement of the question?

Try to graph such a function.
Give it a try.
 
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  • #3
Generally, the "zero-function" is f(x)= 0 for all x so that "non- zero" means "not equal to 0 for at least one x". But we can have f(x)= 0 for x= 1, non-zero for x not equal to 0, and its Riemann integral, [itex]\int_a^b f(x)dx[/itex], from any a to any b, is non-zero. If you use the Lebesque function, then we can have f(x) equal to zero on any set of measure 0 and have [itex]\int_a^b f(x)d\mu[/itex] non-zero.
 
  • #4
following up on maajdl's comment, the existence of such a function f is equivalent to the existence of a strictly increasing or decreasing differentiable function F : R --> R such that F(c^2+1) = F(c).

But, it is easily seen that no such function can exist, because the derivative of F(x^2+1) is equal to the derivative of F(x), which means that 2xf(x^2+1) = f(x). Plugging in x = 0 shows that f(0) = 0. Assuming you mean non-zero as in f can not take the value 0.
 

1. What is an integral problem?

An integral problem is a mathematical problem that involves finding the area under a curve or the accumulation of a quantity over a certain interval. It is typically solved using techniques from calculus.

2. What does the term "non-zero continuous function" mean?

A non-zero continuous function is a mathematical function that is defined and does not equal zero for all values of its independent variable. It is also continuous, meaning that there are no abrupt changes or breaks in the graph of the function.

3. What is the domain and range of a solution to an integral problem?

The domain of a solution to an integral problem is the set of all values of the independent variable for which the integral is defined. The range is the set of all possible values of the dependent variable, which is typically a real number in this case.

4. How is a non-zero continuous function represented in a graph?

A non-zero continuous function is typically represented in a graph by a smooth, unbroken curve. The x-axis represents the independent variable, while the y-axis represents the dependent variable.

5. What are some real-world applications of solving integral problems?

Solving integral problems has many real-world applications, such as calculating the area under a curve to determine the volume of a shape, finding the rate of change of a quantity over time, or determining the displacement of an object over a given interval. It is also used in fields such as physics, engineering, and economics to model and solve various problems.

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