I came across the following problem and I have been cracking my head with it:  Prove or disprove that there are no functions $f$ and $g$ such that   $$f(x,g(y-x)+g(z-x))=y^2+z^2$$  for all $x,y,z \in \mathbb{R}$.  The point is that I never had contact with functional equations before so I don't even have the idea of how to start this.  However I did try some things, but I don't really know what to do with them exactly because I lack theory.  Considering $y=z=\alpha$, leaving $x$ free, this yields:  $$f(x,2g(\alpha-x))=2\alpha^2$$  Also, if I consider $x=y=\beta$, leaving $z$ free, this yields:  $$f(\beta,g(0)+g(z-\beta))=\beta^2+z^2$$  And, if I consider $x=z=\gamma$, leaving $y$ free, this yields:  $$f(\gamma,g(y-\gamma)+g(0))=y^2+\gamma^2$$  Finally, if I consider $x=y=z=\delta$:  $$f(\delta,2g(0))=2\delta^2$$  Could you help? Also, would appreciate some suggestions for basic literature in functional equations.
How do I show that if there are functions $f,g$ such that$$ f(g(x)+g(y))=bx+cy $$holds for all $b,c\in\mathbb{R}$, then we necessarily have $b=c$? Is this even true? It seems so, but I'm just not sure how to explain it in detail and using theory from functional equations. Any ideas?  Extra: In addition, I'd like to prove that there are no pair of functions $f,g$ such that$$ y^2+z^2=f(x,g(y-x)+g(z-x)). $$ Any ideas on how to prove this? Plus, does anyone know some literature regarding this type of problems?
Use Unity API from another Thread or call a function in the main Thread How does a pharmacy discount card work? Why use coherent state path integral? What is its motivation or goal? In almost all textbooks of quantum field theory for high energy, they insert the position and momentum eigenstate to formulate the path integral. While in condensed matter field theory, they insert the coherent state to get the path integral. What's the motivation or goal to use the latter instead of the former? How do I get lightning:datatable to do some undocumented thing X? Canonical Question  I want to do &lt;some really cool thing&gt; with a lightning:datatable, but I can't seem to figure out how to do it. I checked the , but I did not see a way to implement &lt;this really cool feature I need&gt;.  Perhaps I want to provide tooltips, or a pick list, or alter the behavior of the menus, or display images, or show a custom component I created, or define a new row-level click behavior, or alter the keyboard event handlers, etc.  Can I somehow do this? Are universal power supplies for laptops worth it? My laptop's cable broke and sadly it seems like I need to buy a new power supply.  What I wonder is, I've found that universal power supplies are much cheaper than the official HP ones. Are they really worth it?  Another question: the original supply has these specs: 65W, 18.5V, 3.5A. However the universal supplies are more powerful (90W, 100W) and come in integer voltages (17V, 18V, 19V, etc). Are they compatible? Should I set them to 18V or 19V? Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. 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