Marginal rate of technical substitution is a mathematical fiction, there is no continuous substitution among factors in actual production plants, yet it is absolutely necessary to make the theory work and find optimal solutions
I've seen many heterodox economists attack consumer theory as unrealistic, while this is a much easier target
Production theory
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The textbook theory of production is just to derive smooth curves, but you do not need it, you can have a lot of kinks, integer variables etc. Different models have different purposes. If you are interested in realistic and detailed production plans, then use mathematical programming algorithms (linear, integer, mixed or nonlinear, depending on the technology, or dynamic tools). If you ware interested in equilibrium, the problem is used to motivate a nice demand curve that fits well aggregated data. So, the modeler builds whatever suits better the problem at hand.
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"there is no continuous substitution among factors in actual production plants,"
False. There are INSTANCES where that may be true, but your assertion is just as wrong as assuming factors are always continuously substitutable.
And no one is preventing you from writing whatever production function you want. Do it and we shall see what you get.