If a function is not continuous at a point in its domain, one says that it has a discontinuity there. There are different types of discontinuities:
- Removable discontinuity.If f(a) and are defined, but not equal.
- Jump discontinuity or step discontinuity, if the one-sided limit from the positive direction and the one-side limit from the negative direction are defined, but not equal.
- Essential discontinuity or Infinite discontinuity, One or both of the one-sided limits does not exist or is infinite.
Function with a Point Discontinuity (Renovable discontinuity)
- For all x-values except 3, the function is defined by the equation
- Graph y=x+3. At x=3, the line has a "hole" or point discontinuity because, at this single point, has no y-value.
- When x=3, the functions is defined to have a value of 1. Indicate this by graphing the single point (3,1), as shown in the accompanying figure.
Function With a Jump Discontinuity
When x takes on negative values, f is defined by the equation f(x) = -x2+5. When x takes on non-negative values, f is defined by the equation . At the crossover point, x=0, the function changes definition from f(x) = -x2+5 to .
Since these equations do not return the same y-value when x=0, the graph of this function has a vertical jump discontinuity, as shown in the accompanying figure:
Function With Infinite Discontinuity
Discuss the continuity of the function
The given function becomes positively infinite as
and negatively infinite as :
Hence, does not exist. The function has a vertical asymptote with an infinite discontinuity at x=2, as shown in the accompanying figure:
Estudy the continuity on R of the function: