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Please answer the question at the bottom using the template provided: Question also located at the bottom: Prove that if f is locally linear on

Please answer the question at the bottom using the template provided:

Question also located at the bottom:

Prove that if f is locally linear on [a,b] then it is actually a linear function on [a,b]. (Hint: Your proof should follow the template of the preceding argument but the line in bold will need to be modified because a single s will no longer suffices.)

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In this question you will need to read and understand the following definition, proposition and proof. Use this as template to answer the question following it. Definition: A function f defined on the interval [a, b] will be said to be locally constant if for every I ( [a, b] there is some o > 0 and some c, both depending on r, such that if I = a then f(2) = c for all z E [a, a + 6) if I = b then f(2) = c for all z e (b - 6, b] . if a 0 and some c such that . f(z) = cfor all z ( [a, a + 6) in the case that w = a . f(z) = cfor all z E (w - 6, w + 6) in the case that a a because w is an upper bound for S. This contradicts that a = w. To get a contradiction in the second case first note that since w is the least upper bound for S and w - 6 0 and some c such that f(2) = c for all z e (b - 6, b]. Now the argument proceeds as before. Since b is the least upper bound for S and b - 6 0 and some A and B, all three depending on r, such that if x = a then f(2) = Az + Bfor all z ( [a, a + 6) if I = b then f(2) = Az + Bfor all z E (b - 6, b] . if a

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