MATH 147 - Advance Calculus - University of Waterloo

Assignment Task

Question 

1. Let f : R → R be a continuous function that is increasing on the rationals, i.e., for all x, y ∈ Q, if x < y>

2. Identify all vertical and horizontal asymptotes by showing that the relevant limits exist (including one-sided, two-sided, limits at ±∞ and limits that are ±∞). You may use results from lectures or work directly with the formal definition. Also identify all infinite limits at ±∞ and all points of discontinuity, stating the type of discontinuity that occurs.

Also Identify all infinite limits at ±∞ and all points of discontinuity, stating the type of discontinuity that occurs.

3. (a) Prove that if f : R → R is a continuous function that vanishes at infinity, then f is uniformly continuous on R.

(b) Is the conclusion in (a) still valid if lim _x→+∞ f(x) = L+ and lim _x→−∞ f(x) = L− are non-zero real numbers? Give a brief justification or counterexample.

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