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Unformatted text preview: x ) 6 = 0 for all x ∈ IR. b . If a function f is diﬀerentiable, then its derivative f is continuous. c . If a sequence of functions converges pointwise, then it converges uniformly. d . If a function f is integrable then it has at most a ﬁnite number of discontinuities. e . Suppose E ⊆ IR is closed, and z is an accumulation point of a sequence { x k } ∞ k =1 ⊆ E . If a function f : E 7→ IR is continuous at each x k , then f is continuous at z . 5. Suppose f : [ a, b ] 7→ IR is diﬀerentiable at x ∈ ( a, b ). Prove that f is continuous at x . 6. Suppose and f and g are integrable over [ a, b ] and c ∈ IR. Prove that ( f + cg ) is integrable over [ a, b ]. 7. Suppose a sequence { f n } ∞ n =1 of continuous functions deﬁned on a common interval [ a, b ] converges uniformly to a function f : [ a, b ] 7→ IR. Prove that f is continuous on [ a, b ]....
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Calculus

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