Examine the continuity of the function defined byf(x) = \(\begin{cases

1.	Examine the continuity of the function defined byf(x) = \(\begin{cases} \frac{\|x-a\|}{x-a} & \quad x \neq a\\ 1, & \quad x = a\end{cases}\)
Answer» Given function is f\( f(x) =\begin{cases}\frac{1x -a_1} {x-a}, & \quad x\neq a\\1 & \quad x=a\end{cases}\) \( f(x) =\begin{cases}\frac{-(x -a)} {x-a}=-1, & \quad x< a\\1, &\quad x=a\\\frac {x-a}{x-a}=1, & \quad x>a\\\end{cases}\) \( f(x) =\begin{cases}-1 & \quad x< a\\+1 &\quad x=a\\1 & \quad x>a\\\end{cases}\) f(a^-)=-1, f(a⁺⁾⁼¹&f(a)=1 \(\because\) f(a^-) ≠f(a⁺) \(\therefore\) f is not continuous at x=a

Examine the continuity of the function defined byf(x) = \(\begin{cases} \frac{|x-a|}{x-a} & \quad x \neq a\\ 1, & \quad x = a\end{cases}\)

Answer»

Given function is

f\( f(x) =\begin{cases}\frac{1x -a_1} {x-a}, & \quad x\neq a\\1 & \quad x=a\end{cases}\)

\( f(x) =\begin{cases}\frac{-(x -a)} {x-a}=-1, & \quad x< a\\1, &\quad x=a\\\frac {x-a}{x-a}=1, & \quad x>a\\\end{cases}\)

\( f(x) =\begin{cases}-1 & \quad x< a\\+1 &\quad x=a\\1 & \quad x>a\\\end{cases}\)

f(a^-)=-1, f(a⁺⁾⁼¹&f(a)=1

\(\because\) f(a^-) ≠f(a⁺)

\(\therefore\) f is not continuous at x=a