As shown in the following graph, why does function sigmoid(x+y)- sigmoid(x-y) has smooth instead of sharp edges around (0,1) in the contour plot? Could you please explain it both intuitive and mathematically?
2026-03-28 20:57:49.1774731469
how to understand sigmoid(x+y)- sigmoid(x-y)
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I think you confused sigmoid function with signum function. Sigmoid function is a function continuous everywhere. There are many possible sigmoid functions. One example is ${e^x\over e^x+1}$.
https://en.m.wikipedia.org/wiki/Sigmoid_function
So unsurprisingly, the plot is continuous, and smooth, since being continuous is implied by differentiability, which is a common characteristic of smooth functions.
A signum plot on the other hand, would be discontinuous and "sharp" like you predicted: