Is there a mathematical theorem that states the equivalence of curved and straight lines Any curve can be considered to be made of sufficiently small straight lines. What is the name of the theorem which states this fact?
Thank you.
I asked the same question at Hacker News. 
[Edit: in the Hacker News question, the OP included the following link, which I think clarifies his/her intent:
http://musr.physics.ubc.ca/~jess/hr/skept/Math/node10.html
--PLC.]
 A: For many calculations that one might want to perform on a "reasonable" curve (e.g., one coordinatized by functions that are differentiable, or piecewise differentiable/smooth/analytic/nice), taking a finely spaced mesh of points on the curve and computing the result on the polygon joining those points, instead of the curve, will produce an answer close to the one for the curve.  The smaller the spacing, the closer the result will be to the result for the curve.  
Quantities approximable in this way include area enclosed by a curve, arc length, integrals of given functions along the curve, winding number around a point, splines, parametrizations, and others.  Quantities not approximable in this way include curvature, which will be zero on the sides of any polygon used as a substitute for the curve, and integer-valued "global" quantities such as number of tangent lines.
A: "Calculus is the acknowledgment that anything but a linear function is far too complicated for us to handle.  Differentiation provides the techniques to tame functions by making them locally linear, and integration comprises the rules for sticking the local pieces together again." -- Alf van der Poorten, Notes on Fermat's Last Theorem (Lecture IX).
From this point of view, one might nominate the Fundamental Theorem of Calculus (to which van der Poorten is alluding) as an answer to this question.
A: Maybe what you need is the Stone-Weierstrass Theorem, which you can use to show (roughly) that a continuous function on a compact set can be approximated arbitrarily well by continuous piecewise-linear functions. The boolean ring version of the theorem is directly applicable here.
