With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as , softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input as , formula_21 However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The failure to converge uniformly is because for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example,