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7 Cards in this Set
- Front
- Back
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Let f be a real valued function and a ∈ D(f). f is continuous x = a if, |
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Let f be a function and let I be an interval, |
1. If f is continuos at x = a, ∀a ∈ I then f is continuous on I
2. If f is continuos at x = a, ∀a ∈ ℝ then f is continuous on ℝ
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Theorem 1 |
Let f be a real valued function and let a ∈ D(f). Then f is continuous at x = a iff ∀ε>0 ∃δ> 0 s.t |x - a| < δ => |f(x) - f(a)| < ε |
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Theorem 2 |
A polynomial is continuous on ℝ |
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Theorem 3 |
Let f and g be functions which are continuous at x = a. Then, 1. f + g is continuous 2. rf is continuous 3. fg is continuous 4. f/g is coninuous, provided that g(a) ≠ 0 |
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Theorem 4 |
1. f(x) = sin x is continuous at x = a 2. g(x) = cos x is continuous at x = a |
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Intermediate value theorem |
Let a < b If f is continuous on [a,b] with f(a) ≠ f(b). For any r between f(a) and f(b),
there exist x0 in (a,b) s.t f(x0) = r |
