Repost from Brilliant: Can anyone solve this?

"Define $f_a^b(x)$ as a function which converts $x$ into base $a$ and then interprets it as a number in base $b$.

For example, $f_2^{10}(0.5)$ will mean first changing $0.5$ to base $2$ i.e. $0.1$ and then interpreting $0.1$ as a base $10$ number. That's it!

So, find a general formula for $\displaystyle \int_0^1 f_a^b(x)~\mathrm dx$"

Question

Repost from Brilliant: Can anyone solve this?

"Define $f_a^b(x)$ as a function which converts $x$ into base $a$ and then interprets it as a number in base $b$.

For example, $f_2^{10}(0.5)$ will mean first changing $0.5$ to base $2$ i.e. $0.1$ and then interpreting $0.1$ as a base  $10$ number. That's it!

So, find a general formula for $\displaystyle \int_0^1 f_a^b(x)~\mathrm dx$"

geerky42 · Answer

Well, is $f_a^b(x)$ even continuous function?

ganeshie8 · Answer

that doesn't matter for definite integral right

geerky42 · Answer

It doesn't? I'm not sure haha. I'm interesting in solution.

ganeshie8 · Answer

:) recall the definite integrals of famous discontinuous functions, 
for example, greatest integer function

ganeshie8 · Answer

the problem looks perfectly fine to me
somehow if we could represent the integrand as a series..

geerky42 · Answer

Well, what I'm saying is that $f_a^b(x)$ may be discontinuous at any value of x, given that we are converting bases. Hard to imagine "area" under it.

geerky42 · Answer

maybe this problem is too advanced for me lol

geerky42 · Answer

I believe we should start by finding "formula" for $f_a^b(x)$ before evaluating integral.

freckles · Answer

like something like this:
|dw:1441131850218:dw|
@geerky42 
?
this is kinda what I'm seeing too in my mind
maybe the points aren't exactly where they should be or whatever but something like this