Let $f(x)=\begin{cases}
2x + 3&\text{ for }x\geq 1,\\
-x+5 &\text{ for }x<1.
\end{cases}$
$f$ is continuous from the right at $x\geq1$.
The proof would be:
Let $\epsilon>0$ be arbitrary.
Let $x_0\geq1$.
Let $\delta=\epsilon/2$.
Let $x\in R$ and $x_0\leq x<x_0+\delta$. Thus $x\geq1$.
Thus $|f(x)-f(x_0)|=|2x+3-2x_0-3|=|2x-2x_0|=2|x-x_0|<2\delta=2\epsilon/2=\epsilon$.
This comes from the definition for continuity from the right:$\forall\epsilon>0\; \exists\delta>0$ such that if $x\in I$ and $x_0\leq x<x_0+\delta$ then $|f(x)−f(x_0)|<\epsilon$.
Prove $f$ is discontinuous from the left at $x=1$ using the definition: $\exists\epsilon>0\; \forall\delta>0$ such that if $x\in I$ and $x_0-\delta<x\leq\ x_0$ then $|f(x)−f(x_0)|\geq\epsilon$.
I can't seem to find $\epsilon$. But I think the proof would go like:
Let $\epsilon$ = ?
Let $\delta >0$ be arbitrary.
Let $x_0<1$.
Let $x\in I$ that is to say $x<1$ and $x_0-\delta<x\leq\ x_0$
Then from there is figuring out $|f(x)−f(x_0)|\geq\epsilon$ which I don't get because $|f(x)−f(x_0)|=|-x+5 +x_0-5|=|-x+x_0|=|x-x_0|<\delta$.
So would you set $\epsilon\leq\delta$?
I know my definitions are correct. My teacher has drilled them into our brains. $I$ stands for the domain of $f$ which is the reals or $R$ except the domain is split in two. And by "from the right" and "from the left" I mean that the space between $x$ and $x_0$ denoted as $\delta$, or $|x-x_0|<\delta$, is only calculated on one side, either adding or subtracting $\delta$, not by doing both which would be $x_0-\delta<x<x_0+\delta$.
Best Answer
Hint 1: Show that $\lim_{x\rightarrow 1^{-}}\left(f(x)\right)$=4. However $f(1)=5$ which is not equal to the left limit. Showing this is sufficient to say that $f$ is not left continuous at 1. Hope this helps.
Hint 2 : Take $ϵ=1/2$. Let $δ>0$. What I want you to do is to prove the converse of the definition of left continuity. Suppose $x<1$. Observe that $|f(x)-f(1)|=|-x+5-2(1)-3|=|-x|=|x|$. Clearly which ever interval $(1-δ,1)$ that is formed depending on $δ$ there always exists a $x$ that is in $(1-δ,1)$ such that $|x|>1/2$ ($|f(x)-f(1)|>1/2$). This is not written in the most formal way I hope you can figure out the whole answer.