LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.

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Presentation transcript:

LIMITS OF FUNCTIONS

CONTINUITY

Definition (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous. DEFINITION: CONTINUITY OF A FUNCTION

Theorem (p. 113)

Question 8 EXAMPLE Solution: 1. Given the function f defined as, draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.

y x Test for continuity: at x=3 1.f(3) is not defined; since the first condition is not satisfied then f is discontinuous at x=3.

Question 8 2.Given the function f defined as draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous. EXAMPLE

y x Solution:

Question 8 EXAMPLE 3. Given the function f defined as, draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.

HA

Question 8 Solution: y x 1 1

Figure (p. 110) The figure above illustrates the function not defined at x=c, which violates the first condition. The figure above illustrates that the limit coming from the right and left both exist but are not equal, thus the two sided limit does not exist which violates the second condition. This kind of discontinuity is called jump discontinuity.

Figure 1.51 (p. 110) The figure above illustrates that the limit coming from the right and left of c are both, thus the two sided limit does not exist which violates the second condition. This kind of discontinuity is called infinite discontinuity. The figure above illustrates the function defined at c and that the limit coming from the right and left of c both exist thus the two sided limit exist. But which violates the third condition. This kind of discontinuity is called removable discontinuity.

2 y x 4

2 4 y x 3 Removable Discontinuity

2 4 y x