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12 Cards in this Set
- Front
- Back
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Open sets |
An open set is when every point x₀ in U there exists some r>0 such that Dr (x₀) is contained within U; symbolically, we write Dr (x₀) ⊂U |
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Open disk/ball |
An open set in the shape of a circle or sphere |
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Boundary Points |
A point x∈Rⁿ is called a boundary point of A if every neighborhood of x contains at least one point in A and at least one point not in A |
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Limit |
The value that f(x) would approach as it gets closer to x₀ but does not reach the point. If it approaches one point from both sides, it does exist |
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If limit F(x)=b, then the limit of cf(x)= |
cb |
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If limit F(x)=b, and the limit of g(x)= c, then the limit of (f+g)(x)= |
b+c |
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If m=1, and the limit F(x)=b, and g(x) =c, then limit (fg)(x)= |
bc |
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If m=1, and the limit F(x)=b≠0 for all x, then limit 1/f(x)= |
1/b where 1/f is defined |
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If f(x) = (f₁(x),...fm(x)) where fi: A→R, i=1,...,m are the component functions of f, then limit f(x)=___ if and only if limit fi (x) = _____ |
If f(x) = (f₁(x),...fm(x)) where fi: A→R, i=1,...,m are the component functions of f, then limit f(x)=b= (b₁,..., bm) if and only if limit fi (x) = b for each i |
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Continuous function |
A function with no jumps or breaks |
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Properties of continuous function (5) |
1. If f(x) is continuous, so is cf(x) 2. If f(x) and g(x) are continuous, so if f(x) + g(x) 3. If f(x) and g(x) are continuous, so is f(x)g(x) 4. If f(x) is continuous and nowhere zero, then the quotient 1/f is continuous at x₀ 5. If f: A⊂ Rⁿ→R^m and f(x)= (f₁(x),..., fm(x)) then f is continuous at x₀ if and only if each of the real valued functions is continuous at x₀ |
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Continuity of compositions |
Let g: A⊂Rⁿ→R^m and let f: B⊂R^m →R^p. Suppose g(A) ⊂B, so that f⊗g is defined on A. If g is continuous at x₀∈A and f is continuous at y₀=g(x₀), then f⊗g is continuous at x₀ |
