"To a first approximation, floating-point numbers behave like real numbers" This is incorrect. To count as an approximation, there have to be some accuracy bou

"To a first approximation, floating-point numbers behave like real numbers" This is incorrect. To count as an approximation, there have to be some accuracy bounds. This is impossible to define, as the reals don't have the same cardinality as any floating point number system. Now, for many interesting and useful real-valued functions, a float-valued function can be defined that is an approximation. But there is no general way to map a function defined on the reals to a function defined on the floats with a known accuracy.