Question

Examine the continuity of:
$f\left(x\right)=x^2-x+9$ for $x\le 3$
$=4x+3$ for $x>3,atx=3$

Answer 1

$f\left(x\right)=x^2-x+9,x\le 3$

$=4x+3,x>3\to 1$

Checking continuity at $x=3$,

$\to \underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}4(3+h)+3=15\to 2$

$\to \underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}{(3-h)}^2-(3-h)+9=15\to 3$

$\to f\left(3\right)={\left(3\right)}^2-\left(3\right)+9=15\to 4$

From $2,3,4$

$\Rightarrow \underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}f\left(x\right)=f\left(3\right)$

$\Rightarrow f\left(x\right)$ is continuous at $x=3$