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Measure of an open ball with no symmetry
Suppose $f:\mathbb{R}^3 \to \mathbb{R}$ is a continuous function such that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Prove that $f$ is a constant function.
The Hahn–Banach theorem is insufficient because if we fix any $x_0 \in \mathbb{R}^3$ then the collection of functions $f(x_0+x)$ does not separate points.
A:
If $f$ were not constant, $f\ge0$ everywhere. Since $f(x+y)=f(x)+f(y)\ge0$ for all $x,y\in\mathbb{R}$, the set $A=\{x\in\mathbb{R}:f(x)>0\}$ is open.
Now, if $x\in A$, then $$f(x)=\frac{f(x+y)-f(y)}{y}\to f(0)=0$$ as $y\to0$. This contradicts our hypothesis.
When it comes to software developers, the code is written, the proof is done.
A new national survey commissioned by the Microsoft Corporation of some 4,000 developers suggests otherwise.
According to the survey, some 74 percent of developers are still waiting for the „proof of concept” or the real-life production application to be handed off to a customer.
„Getting a customer to buy a product is a notoriously difficult process. Those that are able to demonstrate value to customers are often the ones that go on to become the best-selling products,” said Jeff Skoll, corporate vice president, Platform Marketing, Microsoft.
The study also found that more than a third of surveyed developers are not provided with enough time to design and develop the final version. And the majority of respondents (74 percent) are unhappy with their boss’ way of doing things.
„This report is a call to action for the entire industry to improve
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