Limits Continuity And Differentiability

Consider the function \(f:(0,2) \rightarrow \mathbf{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function \(g(x)\) defined by

$$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x < 2 \end{array} .\right. \text { Then, }$$