Question

Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable.

1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into two parts. • First we show that f is bounded. Since limp-ro f (x) = L, there is a d with 0<8<min {xo - a, b - To} such that \f (x) – L||<1 whenever 0 < |x – Xol<8. Show that \f(x)| <1+|L| whenever 0 < |2 – 2005 . - Explain why M1 = supIf (x): 3 € a, xo – p{1}{2)) : 2 e [u, 30 - 1} {if(a): 2 € [70 +.]} and M2 = sup exist. Conclude that If (x) <1+|L| + \f (x0)| + M1 + M2 for all x € [a, b]. • Now we show that given e > 0, there is a partition P of [a, b] such that U (f, P) – L(f,P) < €. - Since limo- f (x) = L, there is a d with € 0 < d < min 20 – a,b – 20, 3 max{1+ f(Co) - L], 2} such that f (x) - L[<1 whenever 0 < |< – xo <8. Show that 8 8 sup }]}-inf{s xot (2):3€ 20 - -inf{f (x): 2 € 20 – 2:2 .]} <max {1+ \f(x0) – L1,2} Explain why there is a partition P1 of (a, to – ] such that U (f, P1) – L (f, P1) < ş. – Explain why there is a partition P2 of [xo + 9, 6] such that U (f, P2) – L(f, P2) < ş. Consider the partition P = P, U P2 of [a,b], show that U (f, P) – L (f,P) < €.

Answer 1

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