ok, but with this definition of “approximate”, a piecewise linear function with finitely many pieces cannot approximate the Weierstrass function.
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.
The answer is yes.
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.
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.