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For the function: $f(x) = \begin{cases}0 & ~~\text {if}~~ x=-1;\\ -x & ~~\text{if}~~ -1 \lt x \lt 0;\\ x & ~~\text{if } 0 \le x\le 1 \end{cases}$ $f(x+2) = f(x) + 1$ make this graph, and watch the graph.

I'm sure that $y=f(x)$ is discontinuous at $x=1$, but about $x \in [0,1]$, $x$ is continuous at $x=1$ ?

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    Have you drawn the graph? It is a rising sequence of disconnected V shapes.2012-02-17

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Let's sketch the graph of this function first over $[-1,1]$.

It looks messy:

$f(x)= \cases{\color{darkgreen}{0}, & $\color{darkgreen}{x=-1}$ \cr\color{maroon}{ -x }& $\color{maroon}{-1\lt x\lt 0} $\cr \color{darkblue}{x}, & $\color{darkblue}{ {0\le x\lt 1} }$ \cr}$

$f$ s a piecewise defined function. To sketch the graph of $f$, look at each "piece" separately:

When $\color{darkgreen}{x=-1}$, the function has the value $0$; so the point $(-1,0)$ is on the graph. We plot this point in the graph below.

For the range of $x$-values in $\color{maroon}{-1, the rule for $f$ is $\color{maroon}{f(x)=-x}$. Over the interval $(-1,0)$ sketch the graph of $\color{maroon}{y=-x}$. Note that the value of $f$ at $-1$ is 0 and is not given by the rule $f(x)=-x$. The "closed" and "open" dots on the graph below indicate what value $f$ takes at the point $x=0$.

For the range of $x$-values in $\color{darkblue}{0\le x\le1}$, the rule for $f$ is $\color{darkblue}{f(x)= x}$. Over the interval $(0,1)$ sketch the graph of $\color{darkblue}{y=-x}$. Note that the value of $f$ at $ 1$ is 1.

enter image description here

So, that's the graph of $f$ over $[-1,1]$.

What about the rest of the graph? Well, that's obtained from the rule $ f(x+2)=f(x)+1. $

So, for example: $f(3)=f(1+2)=f(2)+1$ and $f(3.5)=f(1.5+2)=f(1.5)+1$. To evaluate $f$ at $f(x+2)$, you just take the value of $f$ at $x$, and add 1.

So, to obtain the graph of $f$ over $[1,3]$, you just draw the same shape as the graph over $[1-2, 3-2]=[-1,1]$ but raise it vertically by 1 unit.

To obtain the graph of $f$ over $[3,5]$, you just draw the same shape as the graph over $[3-2, 5-2]=[1,3]$ but raise it by 1 unit. etc...

Below, is the graph of $f$ over three of its "periods":

enter image description here

Now, looking at the graph, you can see that $f$ discontinuous at every odd integer $x\ge -1$ and that $f$ is continuous for any $x>-1$ that is not an odd integer.

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By the definition , the formula for the function in a right neighborhood of X=1 is : y= -X+3. The lateral limits at X=1 are not equal . Then the function is not continuous at X=1.(is left continuous).