$f: \mathbb{R} \rightarrow \mathbb{R}. f(x) = 0$."take all decimals of x with an even index".

Question

I remember this function from when I tried (and failed) a math study 20 years ago.

The function $f$ is a function from $ \mathbb{R}$ to $ \mathbb{R}$. In base 10, it copies the number before the decimal separator and from after the decimal separator it copies all decimals at even places and concatenates them to form the output number. For example:

$f(555.11223344559697989900) = 555.1234567890$

1. Does this function (or a similar one) have a name? Where can I find more info?

I vaguely remember that this function had some interesting properties, but I cannot reconstruct them.

2. Is this a continuous function? When applying the definition of continuity I find on Wikipedia, it seems continuous.
3. Is this function differentiable? Again, when I apply the definition of differentiability myself it seems differentiable.

Perhaps (2) and (3) were the "interesting properties", as they were somewhat surprising for me.

Answer 1

You need to be a bit careful with your definition, because for example $$ 0.2159999\ldots \qquad\text{and}\qquad 0.216000\ldots $$ which represent the same real number $\frac{27}{125}$, would lead to two different outputs ($0.1999\ldots$ and $0.1000\ldots$) by your description. So it doesn't quite manage to define a function from the real numbers.

There are ways to patch things up to work around this problem, but depending on what you want to use the function for, you might prefer different patching-up tactics, and they'd lead to slightly different functions.

I don't think the function has a nice short name. One could describe it as one of the projections corresponding to Cantor's digit-interlacing pairing function, or something like that. It is most often encountered in connection with an argument that once can define a one-to-one correspondence between $\mathbb R\times\mathbb R$ and $\mathbb R$. (In fancier words, $\mathbb R\times\mathbb R$ and $\mathbb R$ have the same cardinality).

It is not continuous -- as the above example illustrates, there is no hope of $f(x)$ having a limit as $x\to\frac{27}{125}$.

Since it is not continuous, it cannot be differentiable.