ok, but with this definition of “approximate”, a piecewise linear function with finitely many pieces cannot approximate the Weierstrass function.
Furthermore, two nonidentical functions f and g cannot approximate each other. Just choose, for a given x, epsilon less than f(x) and g(x); then no matter how small your neighbourhood is, |f(x) - g(x)| > epsilon.
ok, but with this definition of “approximate”, a piecewise linear function with finitely many pieces cannot approximate the Weierstrass function.
Furthermore, two nonidentical functions f and g cannot approximate each other. Just choose, for a given x, epsilon less than f(x) and g(x); then no matter how small your neighbourhood is, |f(x) - g(x)| > epsilon.
The original question is whether a continuous function can be approximated by a linear function at a small enough scale. The answer is yes.
If you want the error to decrease linearly with scale, then continuous is not sufficient of course.
I think we have just established that the answer is no… for the definition of “approximate” that you gave…
Hum no you haven’t. The approximation depends on the scale of course.