MATH1151 Chapter Notes - Chapter 2: Nested Intervals, Fourier Series, Classification Of Discontinuities
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Suppose that a rn is a closed and bounded set and that f : a r is continuous. Then f attains a maximum value at some point of a. This de nition depends on the de nition of lim x x0 f (x). Consider the following functions f (x) = g(x) = The formal de nition of the limit of a function f as x approaches a real number x0 is similar to the -m de nition of f (x). lim x : definition. Formal de nition: we say that lim x x0 f (x) = l if. In words: no matter how small an error tolerance > 0 you specify, we can specify a little interval around x0, say. I = (x0 , x0 + ) so that error(x) = |f (x) l| < whenever x 2 i and x 6= x0.
