1.4 Continuity
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The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
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The document discusses continuity of functions at a point c. It defines a function as continuous at c if the function is defined at c, the limit exists at c, and the function value equals the limit value. Only functions meeting all three criteria are continuous. Discontinuities can be removable if defining the function value makes it continuous, or nonremovable if no definition works. The document analyzes five functions in pictures to identify which are continuous, discontinuous with removable or nonremovable discontinuities.
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1.4 Continuity
- 1. Continuity
- 2. So the 5 picturesof these are In which below, let’s look at: a. Is the function defined at c ? continuous? b. Does the limit exist at c ? 1 2 a. f(c) is und c a. f(c) is und c b. lim DNE b. lim EXISTS 3 4 5 a. f(c) cis defined a. f(c) c defined is a. f(c) is defined c b. lim DNE b. lim EXISTS b. lim EXISTS
- 3. A function is continuous at c if… • 1. The function is defined at c. • 2. The limit exists at c. • 3.The value of function at c equals the value of limit at c. * All three of these must occur.
- 4. Another way of saying this is that a function is continuous at every point in the interval if you can draw it without lifting your pencil.
- 5. Based on the definition of continuity, which of the functions are continuous? a b discontinuous discontinuous c d e discontinuous discontinuous continuous
- 6. Removable vs. Nonremovable Discontinuities • A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) f(c). • If you can simply “plug up the hole”, the discontinuity is removable. • A discontinuity is nonremovable if there is no way to define the function at a point to make it continuous.
- 7. In the 5 pictures below, let’s now identify which have removable and nonremovable discontinuities. a b nonremovable removable c d e nonremovable removable continuous