# Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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### Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

Let $f: \mathbb{R} \to \mathbb{R}$ $$f(x^2+y)=f(x)+f(y^2)$$ How do I find all functions that fulfill this equation? I tried to just write many equalities but it just doesnt help.
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### Indeterminate equation and functional equation

I was wondering what differences and relations are between indeterminate equation and functional equation? Are they the same concept? Thanks and regards!
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### Functional derivative of $\int \left( \frac{df^2 }{d^2 x} \right)^2 dx$

According to page 7 of the PDF document $$\frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx$$ I would like help proving this statement. Although ...
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### All functions with the property $k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
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### Solving the functional Equation $f(f(x))=f(x)+x$ [duplicate]
Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$