is referred to as the softmax function. The reason is that the effect of exponentiating the values x_1,\ldots,x_n is to exaggerate the differences between them. As a result, \operatorname{softmax}(k,x_1,\ldots,x_n) will return a value close to 0 whenever x_k is significantly less than the maximum of all the values, and will return a value close to 1 when applied to the maximum value, unless it is extremely close to the next-largest value. Thus, the softmax function can be used to construct a weighted average that behaves as a smooth function (which can be conveniently differentiated, etc.) and which approximates the indicator function
