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

[Wicketer]

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.

 

[maajdl]

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.

 

[HallsofIvy]

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, \int_a^b f(x)dx, 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 \int_a^b f(x)d\mu non-zero.

 

[disregardthat]

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.