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.
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.
https://en.wikipedia.org/wiki/Differentiable_function#Differ...
Also this isn't advanced mathematics, this is something most who are interested learns in high-school and I can assure you that most physics professors are aware of this fact.
No, they said what I wrote they said. It was at YouTube, from MIT, from a course in quantum mechanics.
Uh, I do understand this stuff: E.g., given a positive integer n, the set of real numbers R, the usual topology for R^n, a set C a subset of R^n closed in that topology, there exists a function
f: R^n --> R
so that for x in R^n, f(x) is 0 for x in C, f(x) > 0 for x not in C, and f infinitely differentiable. E.g., for n = 1, the result applies for set C a Cantor set or a Cantor set of positive measure. For n > 2, the result applies for a sample path of Brownian motion or the Mandelbrot set.
When I was a grad student, I discovered and proved this result. Later I published the result in JOTA.
Or for example, this famous dialogue between Geoffrey Chew and Arthur Wightman from the 1960s bootstrap fad:
Wightman asked Chew: why assume from the start that the S-matrix was analytic? Why not try to derive it from simpler principles? Chew replied that "everything in physics is smooth". Wightman asked about smooth functions that aren't analytic. Chew thought a moment and replied that there weren't any.
It's really nice! Get to understand a lot of stuff.
Some of the advanced math is really powerful for applications.
And, can look back at the math saw in physics where wondered if the physics profs really understood the math and conclude, right, they didn't or at least not very well.
E.g., recently I saw a physics lecture where they explained more than once that a quantum mechanics wave function was differentiable and "also continuous" as if there was some question, doubt, choice, or chance otherwise. Of COURSE it is continuous! Every differentiable function is continuous!
There's a lot more on why it's nice to understand the advanced math!