
Discontinuities
Discontinuities:
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 onesided limit from the positive direction and the oneside limit from the negative direction are defined, but not equal.
 Essential discontinuity or Infinite discontinuity, One or both of the onesided limits does not exist or is infinite.
Function with a Point Discontinuity (Renovable discontinuity)
 For all xvalues 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 yvalue.
 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) = x^{2}+5. When x takes on nonnegative values, f is defined by the equation . At the crossover point, x=0, the function changes definition from f(x) = x^{2}+5 to .
Since these equations do not return the same yvalue 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 :
and
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:
