Exercise $2$: Determine interval of continuity and use intermediate value theorem

Consider the function

$f(x)=\{\begin{array}{cc}{2}^x,& x<mn>0</mn>\\ 2sin(x)+1,& 0x2\\ x-1,& x>2\end{array}$

a$)$ Identify the $x$ values, if any, at which $f(x)$ is discontinuous. Make sure to do this using the definitions, not a guess or a graph!

b$)$ Notice that $f(-1)=2^{-1}=\frac{1}{2}$ and $f()=2.$ Does the intermediate value theorem guarantee that there will be a solution to $f(x)=\frac{3}{2}$ for some xin$[-1,]?$ Explain your answer. If you need to draw a sketch, do that.

c$)$ Notice that $f()=2$ and $f(7)=6.$ Does the intermediate value theorem guarantee that there will be a solution to $f(x)=4$ for some xin$[,7]?$ Explain your answer. If you need to draw a sketch, do that.