Question

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

2. at x = 0

1/x if x ≤ -1

x² - 2 if x > -1

Answer 1

$f(x)$ is continuous at a point $x=a$ if $f(a)$ is defined and $lim_{x\to a} f(x)=f(a).$

The function

$f(x)=\{\begin{matrix} \frac{1}{x}, & \text{if }x\leq-1, \\ x^2-2, & \text{if }x>-1, \end{matrix}$

is defined at $x=0:$

$f(0)=0^2-2=-2.$

$lim_{x\to 0^-} f(x)=lim_{x\to 0^-} (x^2-2)=0^2-2=-2,$

$lim_{x\to 0^+} f(x)=lim_{x\to 0^+} (x^2-2)=0^2-2=-2=lim_{x\to 0^-} f(x),$

so there exist $lim_{x\to 0} f(x)=-2=f(0)$ and the function is continuous at $x=0.$

We can see the continuity at $x=0$ at the graph of $f(x)$ :

Answer: the function is continuous at x=0.