Section 2 4 Continuity Definition A function f x is continuous at a if and only if that f a is defined lim f x x a lim f x f a This statement implies x a exists as a number and the two are equal A function is continuous on an interval if it is continuous at every a in the interval The graph of a continuous function has no holes gaps or vertical asymptotes You can draw the graph without lifting the pencil Polynomials sinx and cosx and exponential functions are continuous on the whole real line Rational functions and the trigonometric functions tan x sin x cos x 1 sec x are continuous cos x wherever the denominator is nonzero and are not continuous discontinuous wherever the denominator is 0 Left and right hand continuity A function f x is continuous from the left at a if and only if lim x a f x f a Similarly f x is continuous from the right at a if and only if Example f x x f x f a is continuous from the right at 0 We say f is continuous on 0 x Examples f x lim x a 2 2x 5 x 2 9 is continuous except at 3 and 3 f has vertical asymptotes x 3 and x 3 g x x 2 2x 3 x 2 9 is also continuous except at 3 and 3 g has a hole at x 3 and vertical asymptote x 3 What do we mean by a hole The limit as x approaches a exists but is not equal to g a To see this factor the numerator and the denominator x 2 2x 3 x 2 9 lim g x x 3 g 3 2 3 3 1 3 3 x 3 x 1 x 3 x 3 2 x 1 x 3 if x 3 This type of discontinuity is called removable since we only need to define 3 to make it continuous at 3 The discontinuity at 3 is non removable What is a gap A piecewise function has a gap at a if the left piece does not meet the right piece at a This means lim x a f x lim x a f x Since the limit does not exist at a the function is not continuous at a Example 3 x 4 2 f x x 4x x 1 lim f x 3 4 7 x 1 1 x 4 lim f x 1 x 4 x 1 2 1 shows f is not continuous at 1 The graph has a gap there Since 7 is the value of f and is the left hand limit f is left continuous at 1 but it is not right continuous Check x 4 lim x 4 f x 4 2 16 lim x 4 f x 4 4 16 and f 4 16 so f is continuous at 4 Intermediate Value Theorem If a b and f is continuous on the interval a b and f a f b then f assumes every value between f a and f b on the interval a b The reason is We cannot draw a continuous curve from a f a to b f b without crossing every horizontal line y C for C between f a and f b How do we use the Intermediate Value Theorem Example Show that the polynomial p x x 3 5 x 3 has a root in the interval 1 1 p 1 3 p 1 9 so there is some c between 1 and 1 where p c 0
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