Re:Lessons I Wish I Had Learned Before Teaching Differential...
by benlorenzetti on Monday September 04, 2017 @06:16PM
(#15171174)
Perhaps he means from a practical point of view. Signals arising from the types of natural phenomena measured in engineering is always real, contiguous, and limited. An integrating op amp circuit is going to be stable up to the limits of the power supply, but a differentiator is likely unstable and unusable due to noise. Fourier transform and freqiency analysis rely on integration. Feedback loops with delay. Etc.
The class of C(1) functions is quite easy. The class of intergrable functions is much more difficult. All we know is that it is larger. Consider this: to prove a function isn't differentiable, you need only give a single point where the derivative as a limit doesn't converge. To prove a function has no integral, you need to consider all possible partitions of that function's domain. (You also need to specify what exact measure is being used, etc).