University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2024
Views:

72

Pages:

1
Author:

Braeden Boyd

Quick Sheet: Continuity 1. Definition: A function f is continuous at x = a if lim f (x) = f (a) x→a 2. This definition says three things: (i) f (a) can be computed. (ii) x→a lim f (x) exists. (iii) The values in (i) and (ii) are the same. • To show that f is continuous at x = a, show that the three properties above hold. • To show that f is NOT continuous at x = a, show which of the three parts do not hold. 3. Theorem: All of our basic functions are continuous on their domains. That is, all polynomials, rational functions, sine, cosine, tangent (and their inverses), exponential functions, logarithms, root functions are all continuous on their domain. Are ALL functions continuous on their domains? No, but probably all of the functions you are familiar with are. A template function for something not continuous anywhere: The function that takes the value 0 on the rational numbers, 1 on the irrationals. (Property (i) holds, but property (ii) does not). Furthermore, all (finite) sums, differences, products, quotients, and compositions of continuous functions are continuous on their new domains. We use this theorem when we “find where f is continuous”. That is, if f was formed by one of the previous operations on our basic functions, we can restate the problem as “find the domain of f ”. 4. Definition (Continuity on a closed interval): We say that f is continuous on a closed interval [a, b] if f is continuous on (a, b), and lim f (x) = f (a) and x→a+ lim f (x) = f (b) x→b− This is a “global” definition of continuity. The original definition was a “local” definition of continuity. 5. The Intermediate Value Theorem: • The formal statement: If f is continuous on [a, b] and N is a value between f (a) and f (b), then there is a c in [a, b] such that f (c) = N . • The intuitive meaning: The image of a closed interval under a continuous function is a closed interval (that is, if the domain is a closed interval, the range must be a solid interval with no breaks). • The main useage: If f (a) > 0 and f (b) < 0, then there must be a solution to f (x) = 0 for some x in (a, b). (The same holds if f (a) < 0 and f (b) > 0). 1

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