§2.5. Continuity In this section, we use limits to

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Presentation transcript:

§2.5. Continuity In this section, we use limits to formulate the notion of continuity. This can be treated as one application of limits. Continuity at one point Continuity on an interval Intermediate value theorem

II. Continuity on intervals Definition: (1) f is continuous on the open interval (a,b) (or (– ,a), (a, ), (– , )) if f is continuous at every number in the interval. (2) f is continuous on [a,b] if f is continuous at every number in (a,b) and is also continuous from right at a and continuous from left at b

Theorem: Polynomials, rational functions, exponential functions, log functions, root functions, trigonometric functions, and inverse trigonometric functions are continuous at every number in their domains. Comment: (1) Above functions are called basic functions. So any basic function is continuous on its domain. (2) The greatest integer function is not a basic function because it has discontinuities on its domain.

Theorem: (1)Any function created by using composition from the basic functions is continuous on its domain. (2) Any function created using combination from the basic functions is continuous on its domain. Comment: The greatest integer function can not be created using basic functions through composition, or combination because it has discontinuities in its domain.

Comment: Most functions used in this course are continuous on their domains. But we still have some functions that have discontinuities on their domains. For example, the greatest integer function f(x) = [x]. How to identify discontinuous points: Usually, there are only a few points in Df where a discontinuity can occur. A suspicious discontinuity point c comes from: (i) The defining rule for f changes at x = c (ii) substitution of x = c causes division by 0 in f