11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on (a, b]. 'x→ 12. The converse to Problem 11 is also true. If f is uniformly continuous on (a,b], then lim,at f(x) exists. We outline a proof. (a) Suppose f is uniformly continuous on (a,b], and let {x,} be any fixed sequence in (a, b] converging to a. Show that the sequence {f(xn)} has a convergent subsequence. (Hint: Use the Bolzano-Weierstrass Theorem.) (b) Let L be the limit of the convergent subsequence of part (a). Prove, using the uniform continuity of f on (a, b], that lim,x¬a* (x) = L. ||

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.