# What is the continuity of f(t) = 3 - sqrt(9-t^2)

What is the continuity of $f\left(t\right)=3-\sqrt{9-{t}^{2}}$
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Warkallent8
$f\left(t\right)=3-\sqrt{9-{t}^{2}}$ has domain $\left[-3,3\right]$
For a in $\left(-3,3\right)$, $\underset{t\to a}{lim}f\left(t\right)=f\left(a\right)$ because
$\underset{t\to a}{lim}\left(3-\sqrt{9-{t}^{2}}\right)=3-\underset{t\to a}{lim}\sqrt{9-{t}^{2}}$
$=3-\sqrt{\underset{t\to a}{lim}\left(9-{t}^{2}\right)}=3-\sqrt{9-\underset{t\to a}{lim}{t}^{2}}\right)$
$=3-\sqrt{9-{a}^{2}}=f\left(a\right)$
So f is continuous on (−3,3).
Similar reasoning will show that
$\underset{t\to -{3}^{+}}{lim}f\left(t\right)=f\left(-3\right)$ and
$\underset{t\to {3}^{-}}{lim}f\left(t\right)=f\left(3\right)$
So f is continuous on [−3,3].