Mathematics 1560 Lecture Notes - Lecture 3: Classification Of Discontinuities, Intermediate Value Theorem
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Interior point: a function y = f (x) is continuous at an interior point c of its domain if limx c f (x) = f (c). If a function is continuous at all interior points of its domain and the domain is an interval, then the function can be drawn without picking up your pencil. Polynomials, rational functions, and the six trigonometric func- tions are continuous at every point of their domains. Consider the piecewise de ned function f (x) = x if x ( , 0) 0 if x = 0 x2 if x (0, ) The book also gives examples of an in nite discontinuity and an oscillating discontinuity (see page 122). Discuss the discontinuities of f (x) = |x| x and g(x) = int x. If f is continuous at c and g is continuous at f (c), then the composite g f is continuous at c.
