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> Every differentiable function is continuous! Under who's definition? The Heaviside step function is discontinuous, but it's derivative is usually considered
by smaddox 10y ago
> Every differentiable function is continuous!
Under who's definition? The Heaviside step function is discontinuous, but it's derivative is usually considered to be the Kronecker delta function. Both of these are used extensively in physics and engineering.
- zodiac 10y agoDid you mean Dirac delta instead of Kronecker delta?
- smaddox 10y agoYes, my mistake. Kronecker is used often in quantum mechanics, but when working in Hilbert space rather than real space.
- witty_username 10y ago> If f is differentiable at a point x0, then f must also be continuous at x0. https://en.wikipedia.org/wiki/Differentiable_function#Differentiability_and_continuity https://en.wikipedia.org/wiki/Differentiable_function#Differ...
- Smaug123 10y agoThe Dirac delta function is not a function but a distribution. This is just the product of sloppy notation by the physicists.
- Klockan3 10y agoNo, it is just that physicists means something different than mathematicians when they use the word "function". Just assume that they always are distributions that are evaluated with dirac deltas and everything makes sense.
- graycat 10y ago> Under who's definition W. Rudin, Principles of Mathematical Analysis. W. Fleming, Functions of Several Variables. For the set of real numbers R, a function f: R --> R x in R, d in R, function f is differentiable at x with value d provided for h in R lim_{h --> 0} (f(x + h) - f(x))/h = d In this case we write f'(x) = df(x)/dx = d In that case, in particular, lim_{h --> 0} (f(x + h) - f(x)) = 0 so that f is continuous at x. Differentiation is important in physics, e.g., in Newton's second law.