Example 1: Jump Discontinuity
Consider the piecewise function defined as follows:
- For x < 0, y = x + 2
- For x > 0, y = x^2
Sketch a graph of this function. For all x-values less than 0, draw a straight line with slope 1, which intersects the y-axis at y=2. For all x-values greater than 0, draw a parabola opening upwards starting from (0,0). Calculate the left-hand limit as x approaches 0 and the right-hand limit as x approaches 0.
Here, the function has a jump discontinuity at x=0. This violation occurs because the left-hand limit (which is 2) and the right-hand limit (which is 0) at x=0 do not agree and hence the function is discontinuous at x=0.
Example 2: Point Discontinuity
Now consider the function y = x^2, for all x except at x=0.
Draw a graph of a parabola that opens upwards, but at x=0 remove the point. The function is defined for all x-values except for x=0. Here, there is a point discontinuity or removable discontinuity.
This function violates the rule that a function is only continuous if it is defined at a certain point. In the case discussed, the function is undefined at x=0, thereby creating a point of discontinuity.
Example 3: Infinite Discontinuity
Finally, let's consider the function y = 1/x.
Sketch a graph of this function. Remember that the function asymptotically approaches the y-axis (x=0) while never reaching x=0. Either on the positive and negative side of y, as x approaches zero, the function value will move towards positive or negative infinity.
This function shows an example of an infinite discontinuity or non removable discontinuity at x=0. The continuity is violated because as x approaches zero, the function values approach either positive or negative infinity, thereby creating a vertical asymptote.