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The problem is that your assertion is not true for any arbitrary function. For example, consider a function(aka curve) defined by: f(x) is x if x is irrational

The problem is that your assertion is not true for any arbitrary function. For example, consider a function(aka curve) defined by: f(x) is x if x is irrational and 1 if x is rational. This function is discontinuous and cannot be approximated by shorter and shorter straight lines. The calculus concept you are reaching for is continuity. If f(x) is continuous, then for any e>0 there exists a d>0 such that abs( f(x) - f(y) ) < d if abs( x - y) < e. That's the famous epsilon-delta mantra of the calculus. See http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_.... Continuity is a powerful property that enables much of what is useful in the calculus. Continuity of a function and its derivative is even stronger; continuity of the function and first and second derivatives is stronger still.

zeynel1 16y ago
"Consider a function(aka curve)..." Thank you for this answer and I apologize for the ambiguity in my question, especially, regarding the word "curve." By "curve" I meant "curved line" or even more precisely, a "curved line without a gap." Obviously this is an assumption; I ignore if the line is made of points; because I want to compare a curved line with a straight line. So I believe that the question is not about "continuity." (By the way Galileo has a nice discussion about lines and points here http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo%20galilei&pg=PA26#v=onepage&q&f=false http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo...) Maybe I can ask the same question like this. Here are 3 numbers: 0.0872664626; 0.1047197551; 0.1221730476. Is it possible to tell if these are points on a straight line or if they are points on a curved line? I believe both; depending on the scale they are plotted; the numbers would look like a straight line on some scale; but they are points on a sin curve. In other words, nature does not distinguish between straight line and curved line; the same three numbers can be read both ways. So I am looking for a theorem that would describe this fundamental property of nature. Thanks.
- drallison 16y ago
  For x sufficiently small, sin(x) looks like x (x in radians). That's because if you write the Taylor's series expansion of sin(x) and ignore high order terms, you get x. The property you are looking for is continuity.

zeynel1 16y ago

"Consider a function(aka curve)..." Thank you for this answer and I apologize for the ambiguity in my question, especially, regarding the word "curve." By "curve" I meant "curved line" or even more precisely, a "curved line without a gap." Obviously this is an assumption; I ignore if the line is made of points; because I want to compare a curved line with a straight line. So I believe that the question is not about "continuity." (By the way Galileo has a nice discussion about lines and points here http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo%20galilei&pg=PA26#v=onepage&q&f=false http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo...) Maybe I can ask the same question like this. Here are 3 numbers: 0.0872664626; 0.1047197551; 0.1221730476. Is it possible to tell if these are points on a straight line or if they are points on a curved line? I believe both; depending on the scale they are plotted; the numbers would look like a straight line on some scale; but they are points on a sin curve. In other words, nature does not distinguish between straight line and curved line; the same three numbers can be read both ways. So I am looking for a theorem that would describe this fundamental property of nature. Thanks.

drallison 16y ago
For x sufficiently small, sin(x) looks like x (x in radians). That's because if you write the Taylor's series expansion of sin(x) and ignore high order terms, you get x. The property you are looking for is continuity.