# Tagged Questions

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### I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
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### Functions such that $f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$

What are the continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for every $x,y\in \mathbb{R}$ $$f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$$
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### Solve this functional equation [closed]

A function $f: \mathbb R \rightarrow \mathbb C$ is such that $f(0) = 1$, $f(-t) = \overline{f(t)}$ and $\mathrm{Re} f(t) = f(t) \overline{f(t)}$. Solve for $f$. If you don't mind.
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### Find all $f:\mathbb R\to\mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $xf(y)+yf(x)=(x+y)f(x)f(y)$.

Find all $f:\mathbb R\to \mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$ My try: whenever $y=0$, we have $$x\cdot f(0)\cdot(1-f(x))=0$$ ...
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### A Dangerous Function [closed]

Recently, I have started solving many ques on functional equations. But, this ques for me was tough, $f(y)^{x^2}+f(x)^{y^2}=f(y^2)^x+f(x^2)^y$ I've started substituting some values, trying to ...
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### If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function. What I did: ...
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### Functional inequation on $\mathbb{R}$: $f(x+y^2)-f(x)\geq y$

I have the following equation: $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$\forall (x,y)\in\mathbb{R}^2,\ f(x+y^2)-f(x)\geq y$$ f is not necessarily differentiable/continuous/... (In fact, we can prove ...
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### Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0.$$ Can someone give explicit ...
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### ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
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### submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
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### Prove That $f(n+f(n))=n$

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$
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### Suppose a function $f : \mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$.

Suppose a function $f : \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$. Show that (a) $f$ is one-to-one. (b) $f$ cannot be strictly decreasing, and ...
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### Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
### Does $f(\mathbf x _1 + \mathbf c ,…,\mathbf x _n + \mathbf c)=f(\mathbf x _1 ,…,\mathbf x _n)$ imply…
I'm trying to prove the following claim: Let $\mathbf x _1,...,\mathbf x_n\in \mathbf R ^p$ and $f:\mathbf R ^p \times ... \times \mathbf R ^p \ \ \text{(n times!)}\rightarrow \mathbf R.$ Suppose ...