Norbert Wiener in 1926:
"The brilliant work of Heaviside is purely heuristic,
devoid of even the pretense to mathematical rigor. Its
operators apply to electric voltages and currents,
which may be discontinuous and certainly need not be
analytic. For example, the favorite corpus vile on
which he tries out his operators is a function which
vanishes to the left of the origin and is 1 to the
right. This excludes any direct application of the
methods of Pincherle…
Although Heaviside’s developments have not been
justified by the present state of the purely
mathematical theory of operators, there is a great deal
of what we may call experimental evidence of their
validity, and they are very valuable to the electrical
engineers. There are cases, however, where they lead to
ambiguous or contradictory results..."
and what about Gauss? the whole field was working or knew about the problem and each other knew the other researches, we just slapped the Maxwell name for recognition as he explained how the stuff worked (pretty much as Einstein equations weren't is own), and in the centuries became 'ownage' - it's interesting however to see how 'legends' form
For those interested in the history and mathematical development of Maxwell's Equations as we know them today, Paul J. Nahin's book on Heaviside[1] is a must read.
Why don't other basic functions (constant, reciprocal, parabola …) actually have a name as well? I.e. why did Heaviside’s name stick with the step function?
The step function is about techniques for analyzing discontinuous functions. You can create discontinuous functions by multiplying a normal continuous function by the unit step function. Which was at the time a new thing. It took the mathematical community a long time to get comfortable the techniques. Indeed if you take college math they call it the 'unit step function' in order to avoid referring to it as the Heaviside equation, which is what it's called if you take engineering math.
Constants, reciprocals, parabolas have been known since the time of the ancient Greeks or earlier. Classically they are called Conic Sections. There are other types functions that are named after people. Taylor Series and Bessel Functions come to mind (after Brook Taylor and Friedrich Bessel). There are more of course.
He perfected operational calculus for electrical engineering: https://en.wikipedia.org/wiki/Operational_calculus