Final answer:
The statement that an entire function mapping reals to reals and purely imaginary to purely imaginary is necessarily an odd function is false. One needs to verify the symmetry condition f(-x) = -f(x) for all x to confirm if a function is odd.
Step-by-step explanation:
The statement that an entire function f which takes real values to real values and purely imaginary values to purely imaginary values is an odd function, is false. An odd function is defined as a function f where f(-x) = -f(x) for all x in the domain. Simply mapping real numbers to real numbers and purely imaginary numbers to purely imaginary numbers does not guarantee this symmetry.
However, certain properties of odd functions are quite useful, such as the property that the integral over all space of an odd function is zero due to the cancellation of areas above and below the x-axis. Additionally, an odd function times an even function produces another odd function, while the product of two odd functions yields an even function.
To determine whether f is indeed an odd function, one must specifically check the symmetry condition, which is not explicitly provided by the information that f maps reals to reals and imaginary to imaginary.
Therefore answer is b. False.