(i) Let $f$ be continuous on $[ - 1,1 ]$ and differentiable at 0. For $x \neq 0$, define a function $g$ by $g ( x ) = \frac { f ( x ) - f ( 0 ) } { x }$. Can $g ( 0 )$ be defined so that the extended function $g$ is continuous at 0? (ii) For $f$ as in part (i), show that the following limit exists. $$\lim _ { r \rightarrow 0 ^ { + } } \left( \int _ { - 1 } ^ { - r } \frac { f ( x ) } { x } d x + \int _ { r } ^ { 1 } \frac { f ( x ) } { x } d x \right)$$ (iii) Give an example showing that without the hypothesis of $f$ being differentiable at 0, the conclusion in (ii) need not hold.
(i) Let $f$ be continuous on $[ - 1,1 ]$ and differentiable at 0. For $x \neq 0$, define a function $g$ by $g ( x ) = \frac { f ( x ) - f ( 0 ) } { x }$. Can $g ( 0 )$ be defined so that the extended function $g$ is continuous at 0?\\
(ii) For $f$ as in part (i), show that the following limit exists.
$$\lim _ { r \rightarrow 0 ^ { + } } \left( \int _ { - 1 } ^ { - r } \frac { f ( x ) } { x } d x + \int _ { r } ^ { 1 } \frac { f ( x ) } { x } d x \right)$$
(iii) Give an example showing that without the hypothesis of $f$ being differentiable at 0, the conclusion in (ii) need not hold.