Problems
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Prove that f(x)=1/x is continuous for x<0.
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Prove that f(x)=1/x is continous for x>0.
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Prove that x2 is continuous.
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Determine limx→3−10x4+7x2+300.
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If f(x) is continuous and M(x) is the maximum value of f(x) in the closed interval [x,a], then prove that limx→a−M(x)=f(a).
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If f(x) is continuous and m(x) is the minimum value of f(x) in the closed interval [x,a], then prove that limx→a−m(x)=f(a).
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If f(x) is continuous and M(x) is the maximum value of f(x) in the closed interval [a,x], then prove that limx→a+M(x)=f(a).
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If f(x) is continuous and m(x) is the minimum value of f(x) in the closed interval [a,x], then prove that limx→a+m(x)=f(a).
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Prove that if f(x) is continuous and K is a constant, then f(x)+K is a continuous function.
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Prove that if f(x) is continuous on [a,b] and f(a)<0<f(b) or f(b)<0<f(a), then there exists a number K between a and b such that f(K)=0.
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Prove that if f(x) is continuous on [a,b] and f(b)<0<f(a), then there exists a number K between a and b such that f(K)=0.
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Prove that if f(x) is continuous at a then there is an open interval around a for which f(x) is bounded.
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Prove that if f(x) is continuous on [a,b] and f(a)<0<f(b), then there exists a number K between a and b such that f(K)=0.
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Prove that if g(x) is continuous at a and f(x) is continuous at f(g(a)) then the function composition (f∘g)(x) is continuous at a.
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Prove that if f(x) is continuous at a then there is a closed interval around a for which f(x) is bounded.
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Prove that if g(x) is continuous at a and f(x) is continuous at f(g(a)) then the function composition (f∘g)(x) is continuous at a.
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Prove that if f(x) is continuous at a and f(a)≠0 then 1/f(x) is continuous at a.
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Prove that if limh→0f(a+h)=f(a) then f(x) is continuous at a.
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Prove that if f(x) is continuous at a then limh→0f(a+h)=f(a).
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Prove that f(x) is continuous at a if and only if limh→0f(a+h)=f(a).
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Function Limits of Polynomials
