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Piecewise continuous functions can be approximated by differentiable ones. You lose some properties though. As you pointed out the derivatives might not converg
by contravariant 8y ago
Piecewise continuous functions can be approximated by differentiable ones. You lose some properties though. As you pointed out the derivatives might not converge, or even exist, and the limit converges pointwise at best, not uniformly.
- tprice7 8y ago"Piecewise continuous functions can be approximated by differentiable ones." It depends on how you measure the distance between functions. If you are using the uniform norm (as is used in the universal approximation theorem) then this is false.