# Assume that f is a continuous function.If you are supposed to be at your workplace before 9 (since you have to start at 9 sharp),what does the conclusion in part a) tell you in practice?a ) Suppose...

Assume that f is a continuous function.If you are supposed to be at your workplace before 9 (since you have to start at 9 sharp),what does the conclusion in part a) tell you in practice?

a ) Suppose that f is a continuous function and f(8) = 8.75 . Prove that there is "delta" > 0 such that for all x=( x-delta ; x+delta ) , we have f(x)<9

b) Suppose that x denotes the time when you leave home,and f(x) denotes the time when you arrive at your workplace. Assume that f is a continuous function.If you are supposed to be at your workplace before 9 (since you have to start at 9 sharp),what does the conclusion in part a) tell you in practice? Use real-world language for the formulation.

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### 1 Answer

(1) We are given that f(x) is continuous and that f(8)=8.75

A function is continuous at a point if `lim_(x->c)=L,L=f(c)` (The limit exists at x=c and the value of the function at c is the same as the limit at x=c.) Since f(x) is continuous, it is continuous at x=8.

By definition, the limit at x=8 exists implies that for any `epsilon>0` there exists a `delta>0` such that `0<|x-8|<delta => |f(x)-8.75|<epsilon` . In particular, for `epsilon=.25` there exists a `delta>0` such that `0<|x-8|<delta` (or `x-delta<x<x+delta` ) implies that `|f(x)-8.75|<.25` or `8.5<f(x)<9` .

(2) There is some amount of time such that if you leave your home within that time after 8 you will still be at work by 9. (Suppose `delta=.5` ; then if you leave by 8.05 (8:03) you will be certain to be at work by 9.

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