Continuous functions http://www.bri ghtstorm.com/rnath/calculus/limits-and-continuitv/coiitinuous-functions 1) What kind of a function is a continuous function? 2) Can you give an example of a continuous function? 3) What is a domain of a function? Listening. Listen to the video and fill in the missing information. 1) A function is continuous at x=a if......................................... 2) If a function is continuous at....................in its...................., it is a ................function. 3) f(x)=x3 is a......................, therefore it is.................................. 4) g(x)=x-2 is continuous for........................................................ 5) The third example of an exponential function the presenter gives is 6) Logarithmic functions are defined only for.................................. 7) Natural log is..................................................................... Reading Continuous functions Qs. 1) What does the writer say about the behaviour of a reasonable function? 2) What do the graphs of continuous function look like? 3) What does the third definition say? 4) Why is 1/x a continuous function? 5) How are the functions whose domains are closed intervals defined? 6) Can you name some continuous function in trigonometry? 7) Why is the function x3/sin x continuous? 8) What is the property of the composition of continuous functions? latrín FOCUS A UNIT 3 Continuous Functions Imagine that you had the information shown in the table about some function/ What would you expect the output/(J) to be? X 0.9 0.99 0.999 fix) 2.93 2.9954 2.9999997 It would be quite a shock to be told that/(1) is , say, 625. A reasonable function should present no such surprise. The expectation is that /(l) = 3. More generally, we expect the output of a function at the input a to be closely connected with the outputs of the function at inputs that are near a. The functions of interest in calculus usually behave in the expected way; they offer no spectacular gaps or jumps. The graphs of these functions consist of curves or lines, not wildly scattered points. The technical term for these functions is "continuous", which will be defined in this section. The following three definitions express in terms of limits our expectation that/(a) is determined by values of/(jc) for x near a. Definition Continuity from the right at a number a. Assume that/(,t) is defined at a and in some open interval (a, b). Then the function/is continuous at a from the right if lim.v-,„* /(*) =/(o). This means that 1 lim f(x) exists and 2 that limit is/(a). Definition Continuity from the left at a number a. Assume that f(x) is defined at a and in some open interval (c, a). Then the function/is continuous at a from the left if um.r->„-/to ~fia)- This means that 1 lim /(jc) exists and j'->u- 2 that limit is/(a). The next definition applies if the function is defined in some open interval that includes the number a. It essentially combines the first two definitions. Definition Continuity at a number a. Assume that/(x) is defined in some open interval (b,c) that contains the number a. Then the function/ is continuous at a if \\mMIIf(x) =/(a). This means that 1 lim f(x) exists and JC-+(1 2 that limit is/(a). This third definition amounts to asking that the function be continuous both from the right and from the left at a. The following definitions define the notion of "continuous function"; they depend on the type of domain of the function. Definition Continuous function. Let/ be a function whose domain is the x axis or is made up of open intervals. Then/is a continuous function if it is continuous at each number a in its domain. Thus x2 is a continuous function. So is \!x, whose domain consists of the intervals (-oo, 0) and (0, oo). Although this function explodes at 0, this does not prevent it from being a continuous function. The key to being continuous is that the function is continuous at each number in its domain. The number 0 is not in the domain of I fx. Only a slight modification of the definition is necessary to cover functions whose domains involve closed intervals. We will say that a function whose domain is the closed interval [a, b] is continuous if it is continuous at each point in the open interval (a, b), continuous from the right at a, and continuous from the left at b. Thus VI - x1 is continuous on the interval [-1, 1], In a similar spirit, we say that a function with domain [a,oo) is continuous if it is continuous at each point in (a, oo) and continuous from the right at a. Thus x is a continuous function. A similar definition covers functions whose domains are of the form (-oo, b]. Many of the functions met in algebra and trigonometry are continuous. For instance, 2X, jc, sin x, tan x and any polynomial are continuous. So is any rational function (the quotient of two polynomials). Moreover, algebraic combinations of continuous functions are continuous. For example, since x3 and sin x are continuous, so are x' + sin x, jc3 - sin x, and x3 sin x. The function xVsin x, which is not defined when sin x = 0, is continuous on its domain. The following definitions are needed to make these statements general. Definition Sum, difference, product, and quotient of fimctions. Let/and g be two functions. The functions, f+g, f-g, fg, and fig are defined as follows. if+ g) to = /to + &to ft"1 * sn the domains of both / and g. (f~g) to =/to -