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Posted: Tue Sep 14, 2010 10:20 am
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Posted: Sat Sep 18, 2010 2:34 am
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Posted: Sun Sep 19, 2010 8:53 am
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Posted: Tue Sep 21, 2010 3:34 pm
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Posted: Fri Sep 24, 2010 11:35 pm
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In practice, one learns to recognize which functions are continuous, i.e., have limits equal to their value, so that one can simply plug in the limit point into the function. Polynomials are always continuous, so lim_{x->a}[ f(x) ] = f(a) for any polynomial f(x).
Actually proving a function is continuous at a point is not that simple. The definition of continuity at x = a: for all ε>0, there exists δ>0 such that |x-a|<δ implies |f(x)-f(a)|<ε The intuitive idea is that "if you tell me how close to f(a) you want the function to be, I can always tell you how close x has to be to a in order to guarantee it."
So for your example of f(x) = 2x, one sees that for any given ε>0, one can just pick δ = ε/2 (or anything less than ε/2). Then: |x-a|<δ ⇒ a-δ < x < a+δ ⇒ 2a - 2δ < 2x < 2a+2δ ⇒ -ε < f(x) - 2a < ε ⇒ |f(x) - f(a)| < ε
Again, in practice one can prove very general theorems like: (1) If f(x) is continuous, then cf(x) for any constant c. (2) If f(x) and g(x) are continuous, then so is their sum. (3) If f(x) and g(x) are continuous, then so is their product. (4) If f(x) and g(x) are continuous, then so is their quotient whenever g(x) is nonzero. The properties (1-3) directly imply that every polynomial is continuous from continuity of f(x) = x. And once you know that, you no longer have to play the ε-δ game: "aha, it's a polynomial, so it's limit is the same as its value."
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Posted: Mon Oct 25, 2010 3:43 am
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VorpalNeko In practice, one learns to recognize which functions are continuous, i.e., have limits equal to their value, so that one can simply plug in the limit point into the function. Polynomials are always continuous, so lim_{x->a}[ f(x) ] = f(a) for any polynomial f(x). Actually proving a function is continuous at a point is not that simple. The definition of continuity at x = a: for all ε>0, there exists δ>0 such that |x-a|<δ implies |f(x)-f(a)|<ε The intuitive idea is that "if you tell me how close to f(a) you want the function to be, I can always tell you how close x has to be to a in order to guarantee it." So for your example of f(x) = 2x, one sees that for any given ε>0, one can just pick δ = ε/2 (or anything less than ε/2). Then: |x-a|<δ ⇒ a-δ < x < a+δ ⇒ 2a - 2δ < 2x < 2a+2δ ⇒ -ε < f(x) - 2a < ε ⇒ |f(x) - f(a)| < ε Again, in practice one can prove very general theorems like: (1) If f(x) is continuous, then cf(x) for any constant c. (2) If f(x) and g(x) are continuous, then so is their sum. (3) If f(x) and g(x) are continuous, then so is their product. (4) If f(x) and g(x) are continuous, then so is their quotient whenever g(x) is nonzero. The properties (1-3) directly imply that every polynomial is continuous from continuity of f(x) = x. And once you know that, you no longer have to play the ε-δ game: "aha, it's a polynomial, so it's limit is the same as its value."
crying
I wanted to provide a δ, ε definition but I'm not nearly as clear and concise as you are when writing out math.
Clearly you're a grad student who has had to deal with undergraduates.
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