1. ## Help on continuity

Determine the number at which the function is discontinuous, and show why definition 1.8.1 isnt satisfied at this number
Def 1.8.1
1.)$\displaystyle f(a)$ exists
2.)$\displaystyle \lim_{x \to a}$ exists
3.)$\displaystyle \lim_{x \to a} = f(a)$

$\displaystyle 12.)H(x) =$
$\displaystyle 6 + x$if$\displaystyle x \leq -2$
$\displaystyle 2 - x$if$\displaystyle -2 < x \leq 2$
$\displaystyle 2x - 1$if$\displaystyle 2 < x$

sorry and thanks very much for your help

$\displaystyle 12.)H(x) =$
$\displaystyle 6 + x$if$\displaystyle x \leq -2$
$\displaystyle 2 - x$if$\displaystyle -2 < x \leq 2$
$\displaystyle 2x - 1$if$\displaystyle 2 < x$
The only possible points are $\displaystyle x=-2,2$
Check $\displaystyle x=-2$.

$\displaystyle \lim_{x\to -2}H(x)$ what is it?
When we approach from the right and from the left we have respectively,
$\displaystyle 6+(-2)=4$
$\displaystyle 2-(-2)=4$.
Continous!

$\displaystyle \lim_{x\to 2}H(x)$ what is it?
When we appraoch from the right and from the left we have respectively,
$\displaystyle 2-(2)=0$
$\displaystyle 2(2)-1=3$
Thus it is discontinous.