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“Real” Analysis Is a Degenerate Case of Discrete Analysis

There are many ways to divide mathematics into two-culture dichotomies. An important one is the Discrete vs. the Continuous. Until almost the end of the 20th century, the continuous culture was dominant, as can be witnessed by notation. An important family of Banach spaces of continuous functions is denoted by L , with a Capital L, while their discrete analogs are denoted by the lowercase counterpart l. A function of a continuous variable is denoted by f(x), where the continuous output, f , is written at the same level as the continuous input x, but if the input is discrete, then the function is given the derogatory name sequence, and written an, where the continuous output, a, looks down on the discrete input, n.