# Tag Info

Accepted

### Writing a function on $\mathbb{R}$ as a sum of two injections

The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...
Accepted

### "Simple" integral equation

That is a rather tough puzzle (took me two full days) with a rather short solution. The first step is the differential equation Fred already mentioned: $$(1-z^2)H'(z)-(1+z)H(z)+2zH(z^2)=0\,.$$ Now ...

### Writing a function on $\mathbb{R}$ as a sum of two injections

It works at least for (locally) absolutely continuous functions. Such a function is the integral of a locally $L^1$ function. This weak derivative can be written as a sum of a positive and negative ...

### How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$

Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...
In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...