Is there any set of numbers that, when muliplied with anything from \mathbb{N}$, results in an uneven number?

I noticed the (very easy to see) fact that, whenever you multiply any natural number $n$ with anything having a prime factor of $2$, you get an even number. For example: $ 3 \times 2 = 6$, even $5 \times 6 = 30 $, even and so on. It’s easy to see that this is … Read more

When an odd polynomial is a one-one map on \mathbb{R}\mathbb{R}

Let f(x)=c_1x+c_3x^3+c_5x^5+\cdots+c_{2m+1}x^{2m+1}, c_1,c_3,c_5,\ldots,c_{2m+1} \in \mathbb{R}, m \in \mathbb{N}, namely, f is an odd polynomial over \mathbb{R}. When such a polynomial is a one-one map on \mathbb{R}? Is it always one-one? Example: f(x)=x+x^3. If f(a)=f(b) for a,b \in \mathbb{R}, then a+a^3=b+b^3, so (a-b)+(a^3-b^3)=0, and then, (a-b)(1+(a^2+ab+b^2))=0. Therefore, a-b=0 or 1+(a^2+ab+b^2)=0. In the first case a=b and … Read more

Question about proving existence of a function $f$ such that $f \circ f = g$ for an odd function $g$

I have been reading a book on elementary mathematics and have come to a problem where I don’t understand where the solution comes from. The problem goes : “Let $g : \mathbb{R} \rightarrow \mathbb{R}$ be an odd function, such that $g(x) > 0$ whenever $x > 0$. Show that there exists a function $f : … Read more

Are $f(x)=x$ and $f(x)=-x$ the only odd bijective involutions from $\mathbb{R}$ to $\mathbb{R}$?

This is motivated by a question that was posted last night, but was deleted (I think by the author) before any answers to it appeared. I don’t think it has been re-posted since then. What bijective functions $f$ from $\mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(x)=\frac{f^{-1}(x)-f^{-1}(-x)}{2}?$$ I first spotted that the RHS was the “odd part” of … Read more

Continuous even functions closed and dense

Let $C_e([−1,1],R)$ denote the set of even functions in $C([−1,1],R)$ (a) Show $C_e$ is closed and not dense in $C$. (b) show the even polynomials are dense in $C_e$, but not in $C$. Answer To show that $C_e$ is closed in $C$: We need to show that if we have a convergent sequence of even … Read more

F(x)=∫2−2dyf(x,y)F(x)=\int_{-2}^{2} dy f(x,y) is an even function, is G(x)=∫2−2dy[f(x,y)]2G(x)=\int_{-2}^{2} dy [f(x,y)]^2 even?

I have a real valued function in two real variables f(x,y) which is essentially a black box. The only thing I really know is that F(x)=∫2−2f(x,y)dy is even and that ∫∞−∞F(x)dx=1 Can I conclude that G(x)=∫2−2[f(x,y)]2dy is also even? I looked for a counterexample, but couldn’t find one. I intuitively feel like this should be … Read more

Why does this double integral have the value zero?

Why does the following double integral have the value zero (no calculation needed) ∬ where D={x^2+y^2 \le 2, x \ge0}. Does this have anything to do with symmetry, odd and even functions? Can anyone help me with how to think and understand integrals more, especially how to use symmetry to facilitate integral calculations. Any useful … Read more

What does it mean for a function to be oddly and evenly symmetric?

From understanding, a function describes a relationship between multiple variables, and has unique values across all possible values on one axis while not duplicating violating a vertical line test to determine if values get repeated, and can also potentially be symmetric. Symmetry, from what I understand, is a property of a function that exists in … Read more

Construct a function ff with f′−aff’-af is odd (a>0)

Let a>0, I am trying to construct a function f such that f′−af is odd. i.e f′(−x)−af(−x)=−f′(x)+af(x) By direct computation, we have ddx(f(x)−f(−x))+a(f(x)−f(−x))=0 Solving the ODE, I pick f(x)−f(−x)=−eax. i.e f(x)=−eax+f(−x) But I got stuck here. Any help would be appreciated. Answer Suppose that f(x)=∞∑k=0ckxk Then we want the coefficients of the terms with even … Read more