## FANDOM

10,837 Pages

Call $$f:\mathbb{R} \mapsto \mathbb{R}$$ a CAMFE function (Constant, Addition, Multiplication, Floor, Exponentiation) iff it is of one of the following forms:

• $$f(x) = x$$
• $$f(x) = c$$ for some rational constant $$c$$.
• $$f(x) = g(x) + h(x)$$ where $$g$$ and $$h$$ are CAMFE functions.
• $$f(x) = g(x) \cdot h(x)$$ where $$g$$ and $$h$$ are CAMFE functions.
• $$f(x) = \lfloor g(x) \rfloor$$ where $$g$$ is a CAMFE function, and $$\lfloor x \rfloor$$ denotes the greatest integer less than $$x$$.
• $$f(x) = e^{g(x)}$$ where $$g$$ is a CAMFE function.

Call $$f:\mathbb{R} \mapsto \mathbb{R}$$ finite integral piecewise constant iff all the following are true:

• $$f$$ is piecewise linear with has finitely many pieces.
• $$f$$ is not constant (to weed out trivial cases).
• Each piece is a constant function, and the constants are all integers.
• The boundaries of each piece are at integers.
• The domain of each piece is closed, half-open, or open.
• The domains are disjoint and completely cover $$\mathbb{R}$$.
• The domains have nonzero measure.

Find a CAMFE function that is finite integral piecewise constant.

### CAMFS

What if you replace $$e^{g(x)}$$ with $$\sin(g(x))$$?

### CAMF

What if you completely remove the exponential case?

I presented these problems to Wojowu and he has solved the first two. We're still puzzling over the third, however.