CONTINUITY AND ONE-SIDED LIMITS

1. CONTINUITY AND ONE-SIDED LIMITS • Continuity at a point and on an open interval • Properties of Connectivity • One-sided limits and continuity on a closed interval • The Intermediate Value Theorem

2. Continuity at a point A function, f, is continuous at point, x = c, if there is no interruption of the graph of f at c. • f(c) is defined • exists f is continuous at c, if three conditions are met: The function is defined at c. The limit of the function exists at c. The value of the function at c and limit of the function at c are the same.

3. 4 4 4 -5 -5 -5 5 5 5 dne 1. f(c) is not defined 2. 3. -4 -4 -4 Discontinuity at a Point If f is not continuous at c, then it is said to be discontinuous at c. This can occur if any of the 3 conditions of continuity is not met: Examples 1 & 3 are removable discontinuities because f can be made continuous by appropriately defining or redefining f(c).[These are point discontinuities; limit at c exists] Example 2 is an example of a non-removable discontinuity. [These occur at “jumps”, “gaps” or at vertical asymptotes; limit at c dne.]

4. Continuity on an Open Interval A function is continuous on an open interval, (a, b), if it is continuous at each point in the interval. Why is an open interval specified? Part of the definition of continuity at a point involves finding the limit at that point. To do this, the point must be approachable on both sides. This is not possible for points at the edges of a closed interval. Polynomial, rational, radical and trig functions are continuous at every point in their domains (so are exponential & logarithmic fcns). A function that is continuous on the entire real line, , is everywhere continuous. Examples of such functions are all polynomials, exponential functions, and the sin(x) and cos(x) functions.

5. Continuity on an Open Interval • When asked to discuss the continuity of a function or expression over an open interval you must • Identify any locations of discontinuity • Categorize those discontinuities as removable or nonremovable • Define all intervals of continuity

6. Discuss the Continuity of each function This rational function can not be simplified and has a vertical asymptote at x =0. f has a non-removable discontinuity at x = 0. f is con- tinuous over the intervals (−∞, 0) and (0, ∞). This polynomial function is everywhere continuous since it is defined and continuous over all real numbers. This rational function can be simplified and pro-duces a line, f(x) = x + 1 with a hole at (1,2).f has a removable discontinuity at x = 1. f is continuous over the intervals (−∞, 1) and (1, ∞). This piece-wise function is defined over all reals. At x = 0, the function approaches 1 from the left and from the right. Also, f(0) = 1. Therefore, this function is everywhere continuous. tan x is undefined at x = π/2 + nπ, where it has nonremovable discontinuties. It is continuous on the open intervals:

7. Properties of Continuity If b is a real number, f and g are continuous at c, and h is continuous at g(c), then the following functions are continuous at c: • Scalar multiplication: bf • Sum or difference: f ± g • Product: fg • Quotient: f/g, if g(c) ≠ 0 • Composition: (h ◦ g)(c) or h(g(c)) As a result, the following functions are continuous at every point in their domains: f(x) = x + sin x f(x) = 3 tan x f(x) = sin 4x f(x) = tan (1/x)

8. One-Sided Limits A one-sided limit means that x approaches c from one direction. The limit from the right means that x approaches c from values greater than c and is represented by: The limit from the left means that x approaches c from values less than c and is represented by: Recall that if and only if and . That said, one-sided limits allow us to extend the definition of continuity to closed intervals: A function is continuous on [a,b] if it is continuous on the open interval (a,b) and if and .

9. The greatest integer function: 4 -5 5 -4 dne Discuss the Continuity of This function returns the greatest integer less than or equal to x. The limit as x approaches 0 from the left is The limit as x approaches 0 from the right is The limit as x approaches 0 does not exist, since the left and right sided limits are not the same. Since , the function has a nonremovable discontinuity at x = 0. By similar reasoning, this function has nonremovable discontinuities at every integer and is continuous on the intervals …, [-1, 0), [0, 1), [1, 2), …

10. Discuss the Continuity of The domain of f is [2, ∞). f is continuous over the open interval, (2, ∞). Also, , so f is continuous from the right. Thus, f is continuous over the closed interval, [2, ∞).

11. The Intermediate Value Theorem Pertains to functions that are continuous over a closed Interval. If f is continuous on [a, b], then as x takes on all values between a and b, f(x) must take on all values between f(a) and f(b). Formal theorem: If f is continuous on the closed interval [a, b] and d is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = d. This theorem is often used to find the zeros of a function.If f is continuous on [a, b] and f(a) and f(b) differ in sign, then the Intermediate Value Theorem guarantees that f has at least one zero in the closed interval [a, b].

12. Intermediate Value Theorem Examples • Use Intermediate Value Theorem to verify that the function takes on the indicated value in the indicatedinterval. Then use a Graphing Calculator to estimatethe value of c accurate to two decimal places: • f(x) = x3 + 3x – 2 f(c) = 0 on [0, 1] • f(x) = x3 – x2 + x – 2 f(c) = 4 on [0, 3]

13. Continuity & One-Sided Limits: Summary • Three conditions must be met in order for a function to be continuous at a point, c • The function must be defined at c • The limit must be defined at c • The two values above must be the same • A function can be continuous over an open interval. If that interval is the entire number line, then the function is everywhere continuous. • Limits can be evaluated on one side of a point, extending our definition of continuity to closed intervals. • The continuity of a function is preserved through the operations of scalar multiplication, and function addition, subtraction, multiplication, division, and composition. • The Intermediate Value Theorem is an existence theorem that indicates that a given function value must be included if a function is continuous within a closed interval.