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Question: figure 2 lustration of the inclusion in 34 a show...

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Figure 2: lustration of the inclusion in (34) (a) Show that f() is continuous at E if and only if every sequence rthat satisfies lim also satisfies lim f(zk) = f(z) (b) Show that every polynomial p() (see Exercise 1.5) is continuous at every point (c) Suppose p(-) and q(-) are polynomials unthq(z)メ0, show that |-→ is continuous at (d) Consider the function p q(x) 2 sin ( fz0. Using the -5 version (33) of the definition of continuity, show that f() is continuous at 0 Given ε > 0, what is the largest value of δ that unil uork? One can also define when a function f() is one-sided continuous at a point T. For example, f() is continuous from the left at provided ve >0, δ > 0 so that i-5<1 i If(z) _ f(z)| < ε In this way with-oo < a < b < oo, we can say that f(.) : [a,히 → R is continuous on the interval a, b provided it is continuous at each point E (a, b), continuous from the right at a, and continuous from the left at b.

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