Question

Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-), if x 70 if x = 0 and g(x) = ra sin(1), if r 70 if r=0. (a) Prove that f'(0) does not exist. (b) Prove that g(2) is differentiable on R, and g' (2) is not continuous at r=0. Problem 3: Let f(x) = x2 sin(1), if x is an irrational number if x is a rational number Show that f'(0 exists.

Answer 1

les fico.i] IR fia = HI Clearly too is continuous and bounded but Jenivative ut Heo does not enist.

1 let fiti = Her in Lo.it glal = 1-141 un lo.is thenis fiu- g1H) = ldi - 110 101) RADE- I on Lo, is hince hits is costes difte dentical the Lo, is back heither tits not glans is differentiable on 20.1). Alovi les from = 't Hy on loin gimi = ldil on lois Rutl=flm-gla) = on loin Clearly hit is differentiable on lo, is but neither tex nvt glas is differentiabli { Hisinde it to to colti turcing the granted that the Jeonative of simil is It dito, and obteun.. f'(H) = Sindi - 'cosan At H=o. He lim. I it is not defined. + v) does not exist that mens now. t'ol = lim fro+h)-ful ht

At h-so. His does not tend to any Limit, so that f'lo) does not exist. D 9141) - H? Sind it dito it H = 0. obtain g'(+) = 2018ins - cost (HHO) AR H=0, we appead the definition, and 1947- $cos y = lehel siti (1+0) Detting too, we see that Thu gran is sitte denticble at all pointe di but g'on is not a continuous function, since cost does not exist it Hyo.

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