Consider the graphed piecewise function $f(x)$ shown below.

Determine, for the values $x=-3,x=1,$ and $x=2,$ for which values $f(x)$ is:

Continuous;

Left$-$continuous;

Right$-$continuous.

For each value that $f(x)$ has a discontinuity, state:

Which type of discontinuity it is;

Which of the three criteria for continuity are and are not met;

Why the limit criterion specifically is or is not met.

Determine the open intervals on which $f(x)$ is continuous.

Determine whether $f(x)$ has one or more horizontal asymptotes, using limits at $$ and $-$ in your explanation.Now consider the equation for the function above:

Use the equation and the limit definitions of one$-$sided and two$-$sided continuity to verify the conclusions you drew from the graph regarding continuity and discontinuity at each point, then use the limits at positive and negative infinity to verify the end behavior of the function. Make sure you address each of questions #$1-4$ again here!