##### Example on Continuity

Which one of the following functions is continuous at x = 3 ?

(A)$f(x) = \left\{ \begin{array}{ll} 2 & x = 3 \\ x -1 & x > 3\\ \frac{x+3}{3} & x < 3 \\ \end{array} \right.$

(B) $f(x) = \left\{ \begin{array}{ll} 4 & x = 3 \\ 8-x & x \neq 3\\ \end{array} \right.$

(C) $f(x) = \left\{ \begin{array}{ll} x+3 & x \leq 3 \\ x -4 & x > 3\\ \end{array} \right.$

(D) $f(x) = \left\{ \begin{array}{ll} \frac{1}{x^3 - 27} & x \neq 3 \\ \end{array} \right.$

Things you need to know

For a function to be continuous at 3, the Left Hand Limit (LHL) = Right Hand Limit = $f(3)$

(a)

$f(3) = 2$

LHL = $\lim_{h \to 0} f(3 - h)$  = $\lim_{h \to 0} {\frac{3-h+3}{3}}$ = 2

RHL = $\lim_{h \to 0} f(3 + h)$ = $\lim_{h \to 0} {3 + h -1}$ = 2

∵ LHL = RHL = $f(3)$

∴ continuous at x=3.

(b)

$f(3) = 4$

LHL = $\lim_{h \to 0} f(3 - h)$ = $\lim_{h \to 0} {8 - (3 - h)}$= 5

RHL = $\lim_{h \to 0} f(3 + h)$ = $\lim_{h \to 0} {8 - (3 + h)}$  = 5

∵ LHL = RHL $\neq f(3)$

∴ Not continuous at x = 3

(c)

$f(3) = 3 + 3 = 6$

LHL = $\lim_{h \to 0} f(3 - h)$ = $\lim_{h \to 0} {3-h+3} = 6$

RHL = $\lim_{h \to 0} f(3 + h)$ = $\lim_{h \to 0}{3 + h - 4} = -1$

∵ $f(3) = LHL \neq RHL$

∴ Not continuous at x = 3.

(d)

$f(3)$ is not defined , therefore not continuous at x = 3