Though for this specific discussion (which illustrates the uncountably infinite nature of the set of real numbers), it is useful to note that the class of realizable computers includes both analog and digital variants, though usually we are more familiar with the digital in the form of our desktop/laptop computers or servers.

Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain.

Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun...

Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption.

Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf

And the comment it was from: https://news.ycombinator.com/item?id=11728186