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Maths Exercices
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 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 :

  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:
The fuction is is continuous is discontinuous