Suppose that y=f(x) is defined for all x. Sketch the graph that meets the conditions listed for each of the following.

Question

1. F is discontinuous at x=1 with lim x=-1f(x) = -1 and lim x=2 f(x)=4
2. F is only discontinuous at x=2 and lim x=0= 1/2
3. F is discontinuous at x=0, with limx=2+ f(x)=2 and limx=2- f(x)=-1

Answer 1

(1) plot (-1,-1) and (2,4) on the graph;

draw a straight line from (-1,-1) to (1,0)

draw the DASHED vertical line at x=1

draw horizontal line from (1,4) to (2,4)

This function is f(x) = { (1/2)x - 1/2 if x<=1

4 if x>1

it is discontinuous at x=1 with the specified limits

2) An example function is y = 1/(2-x)

Since the discontinuity is at x=2, the denominator must be x-2

So y = c/(x-2) for some constant c.

Per the given limit, (0,1/2) is on the graph. So

1/2 = c/(-2) ---> c = -1

Hence the function

3) Another piecewise function will do the trick

f(x) = { 0 if x<0

-1 of 0 < x < 2

+2 if x>2

}

You can graph these on desmos but I have uploaded VERY ROUGH sketches for you

in the RESOURCES section under Toolkit. The file name is discontinuous functions.jpeg