$(4$ points$)$ Suppose $f$ is a function such that $\underset{x{2}^{-}}{lim}f(x)=3,\underset{x{2}^{+}}{lim}f(x)=1,$ and $f(2)=3.$

$($a$)$ Is $f$ left continuous at $x=2?$

$($b$)$ Is $f$ right continuous at $x=2?$

$($c$)$ Is $f$ continuous at $x=2?$

$($d$)$ Sketch a possible graph of $f.$

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Activity: $1\mathrm{.}4$ Continuity and Its Consequences

$4.(7$ points$)$ Circle exactly one phrase per box below to complete the statement of the Intermediate Value Theorem.

$5.(4$ points$)$ Let $f(x)=x^3-3x^2+1.$

$($a$)$ Explain why the the Intermediate Value Theorem can be applied to $f$ on the closed interval $1,3.$

$($b$)$ Then use the Intermediate Value Theorem to show $f$ has a root in the open interval $(1,3).$