# Problem 4 (15 marks). Consider a continuous, real—valued function deﬁned on R. Suppose that the value of this function evaluated at some point, say i E R, is positive. This

Problem 4 (15 marks). Consider a continuous, real—valued function deﬁned on R.

Suppose that the value of this function evaluated at some point, say i E R, is positive.

This value can be arbitrarily small, but is strictly above zero. Intuitively, because of

continuity, we expect the function to attain strictly positive values as long as we

restrict ourselves to some small interval around f. Right? (a) Answer whether or not this intuition is correct: YES if you think the intuition is

correct, NO otherwise. (3 marks} (b) If you answered YES to part (a), write down a proposition that formally captures

the intuition. On the other hand, if you answered NO to part (a), explain your

answer as carefully as possible. (6 marks} (c) If you answered YES to part (a), write a proof for the proposition you provided

in part (b). If you answered NO to part (a), construct a suitable counterexample

to illustrate your point. (6 marks}

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