2. Below is the definition of continuity, which you'll see in lecture soon or have already seen in lecture. Part (a) below uses this definition, but the strategy involved in proving (a) is something that we saw within a theorem on limits already. Suppose E C R, f: ER, and let To E E. Then f is continuous at ro iff for each e > 0 there is a > 0 such that if |xo|< 8 and x = E then f(x) - f(xo)| < €. (a) Suppose ECR and f: E → R and To E E be an is an accumulation point of E (note that this is not required in the definition, unlike in the definition of limit). Prove that f is continuous at xo if and only if for any sequence {n}n= that converges to to with In EE for each n, the sequence {f(x)}= converges to f(xo). function (not necessarily continuous) f: ER and xo EE and xo is not an accumulation

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