Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then (1) g is continuous but not differentiable at $\mathrm { x } = 1$ (2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$ (3) g is neither continuous nor differentiable at $\mathrm { x } = 1$ (4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$
Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then\\
(1) g is continuous but not differentiable at $\mathrm { x } = 1$\\
(2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$\\
(3) g is neither continuous nor differentiable at $\mathrm { x } = 1$\\
(4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$