User:MCPOliseno /Math710 CHAPTER 5

1 Let $$ f \ $$ be the function defined by f(0)=0 and f(x)=xsin(1/x) for x $$ \ne \ $$ 0. Find $$ D^+ \ $$ f(0), $$ D_+ \ $$ f(0), $$ D^- \ $$ f(0), $$ D_- \ $$ f(0).

$$ D^+ \ $$ f(0) = $$ \overline{lim}_{h \to 0^+} \frac{f(0+h)-f(0)}{h} \ $$ =$$ \overline{lim}_{h \to 0^+} \frac{f(h)}{h} \ $$ = $$ \overline{lim}_{h \to 0^+} \frac{hsin(1/h))}{h} = \overline{lim}_{h \to 0^+} sin (1/h) \ $$ = 1

3(a) If $$ f \ $$ is continuous on [a, b] and assumes a local maximum at c $$ \in \ $$ (a, b), then $$ D_- f(c) \le D^- f(c) \le 0 \le D_+ f(c) \le D^+ f(c) \ $$.

8 (a) Show that if $$ a \le c \le b \ $$, then $$ T_{a}^{b} \ $$ = $$ T_{a}^{c} + T_{c}^{b} \ $$ and that hence $$ T_{a}^{c} \le T_{a}^{b} \ $$.

(b) Show that $$ T_{a}^{b} (f+g) \ $$ $$ \le T_{a}^{b} (f) \ $$ + $$ T_{a}^{b} (g) \ $$.

11 Let $$ f \ $$ be of bounded variation on [a, b]. Show that $$ \int_{a}^{b} |f'| \le T_{a}^{b}(f) \ $$.

14 (a) Show that the sum and difference of two absolutely continuous functions are also absolutely continuous.

(b) Show that the product of two absolutely continuous functions is absolutely continuous. [Hint: Make use of the fact that they are bounded.]

15 The Cantor ternary function is continuous and monotone but not absolutely continuous.

18 Let $$ g \ $$ be an absolutely continuous monotone function on [0, 1] and $$ E \ $$ a set of measure zero. Then $$ g[E] \ $$ has measure zero.