Let I be an open interval that contrains the point c, let f and g be functions t

Question

Let I be an open interval that contrains the point c, let f and g be functions that are defined on I except possibly at c, and suppose that g is a bounded function. a) Suppose that lim x>c f(x) = 0. Prove that lim x>c (fg)(x) = 0. b) Suppose that f is a bounded function and that neither lim x>c f(x) nor lim x>c g(x) exist. What's an example to show that lim x>c (fg)(x) may still exist.

Explanation / Answer

We say that a function f is continuous from the left at a point c if lim x!c- f(x) = f(c). We say that a function f is continuous from the right at a point c if lim x!c+ f(x) = f(c). Simply to say that a function f is continuous, without specifying some particular point, means that the function is continuous, in the proper sense, at all points where it is defined. Here “in the proper sense” means, for example, that if f is defined only on a closed interval [a, b], then we cannot ask for continuity at a or b, since it is possible to discuss only onesided limits at these points, but it is possible to inquire about continuity from the right at a and continuity from the left at b. Definition We say a function f is continuous on the open interval (a, b) if f is continuous at every point in (a, b). We say f is continuous on the closed interval [a, b] if f is continuous on (a, b), continuous from the right at a, and continuous form the left at b. In the previous section we saw that if f and g are polynomials and c is a point with g(c) 6= 0, then lim x!c f(x) g(x) = f(c) g(c) . 1 Copyright c by Dan Sloughter 2000 2 Continuous Functions Section 2.4 The following proposition restates this fact in terms of our new definitions. Proposition If h is a rational function and h is defined at the point c, then h is continuous at c. In particular, if h is a polynomial, then h is continuous on the entire real line (-1,1). This theorem gives us a very large class of functions which we know to be continuous. a) given: lim x>c f(x) = 0 lim x>c (fg)(x) since g is a bounded function and it doesn't exist on at x=c hence it becomes the form of f(x) and lim x>c (fg)(x) will become lim x>c f(x) = 0

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