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.)

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