Observe the graph of  :

The function  is a piecewise function.

I/D test :

If  on the interval, then  is increasing on the interval.

If  on the interval, then  is decreasing on the interval.

Observe the graph.

 over the intervals ,  and .

Therefore, the function  is increasing over the intervals ,  and .

The function  in the interval .

Since ,  in the interval , where  is a constant.

The function  in the interval .

Since ,  in the interval , where  is a constant.

The function  in the interval .

Since ,  in the interval , where  is a constant.

Here constant  for the continuity of .  (Since  is continuous in )

Graph :

Graph the function  such that above all the conditions are satisfied :

Since ,  in the interval .

Since ,  in the interval .

Since ,  in the interval .

Therefore, the function  exist everywhere except .

Graph :

Graph the function  such that above all the conditions are satisfied :

Graph of the function  :

Graph of the function  :

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