Point of Discontinuity for y=|x|/x: Is it x=0?

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In summary, the function y=|x|/x is a jump discontinuity and the point of discontinuity is x=0, as shown in the graph. The issue of ##\frac00## may be causing confusion or a misconception about the location of the discontinuity.
  • #1
grace77
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Suppose y=|x|/x

I know it is a jump discontinuity but how is the point of discontinuity x=0??
ImageUploadedByPhysics Forums1392327443.441033.jpg
 
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  • #2
$$\lim_{x\rightarrow 0^-}\frac{| x |}{x} \ne \lim_{x\rightarrow 0^+}\frac{| x |}{x}$$
 
  • #3
grace77 said:
Suppose y=|x|/x

I know it is a jump discontinuity but how is the point of discontinuity x=0??
View attachment 66595
Your graph clearly shows that this function is discontinuous at x = 0.
 
  • #4
grace77 said:
Suppose y=|x|/x

I know it is a jump discontinuity but how is the point of discontinuity x=0??
View attachment 66595

What bothers you about that? Do you think the discontinuity should be somewhere else? Are you bothered about ##\frac00##? Answers to these might help uncover your misconception.
 

FAQ: Point of Discontinuity for y=|x|/x: Is it x=0?

1. What is a point of discontinuity?

A point of discontinuity is a point on a graph where the function is not defined or experiences a sudden jump in value. It is a break or interruption in the continuity of a function.

2. How do you find the point of discontinuity for y=|x|/x?

The point of discontinuity for y=|x|/x can be found by setting the denominator (x) equal to 0 and solving for x. In this case, x=0 is the point of discontinuity.

3. Why is x=0 the point of discontinuity for y=|x|/x?

Since the absolute value function is defined as the positive value of the input, when x=0, the denominator becomes 0 and the function becomes undefined. Therefore, x=0 is the point of discontinuity for y=|x|/x.

4. Can the point of discontinuity for y=|x|/x be removed?

No, the point of discontinuity cannot be removed as it is a fundamental characteristic of the function. However, the function can be modified to remove the point of discontinuity by defining it separately for x<0 and x>0.

5. How does the point of discontinuity for y=|x|/x affect the graph of the function?

The point of discontinuity for y=|x|/x at x=0 causes a break in the graph, resulting in two separate branches for x<0 and x>0. This creates a V-shaped graph with an open circle at the point of discontinuity.

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