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Question regarding the continuity of functions
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[QUOTE="pasmith, post: 4499076, member: 415692"] [tex] \lim_{x \to a} f(x) = f(a) [/tex] The domain of a function is part of its definition. A function is defined at all points of its domain and is undefined everywhere else. A function can't be continuous at a point outside its domain because its value is not defined there. The formal definition of continuity of a real function is that [itex]f: U \subset \mathbb{R} \to \mathbb{R}[/itex] is continuous at [itex]a \in U[/itex] if and only if for all [itex]\epsilon > 0[/itex] there exists [itex]\delta > 0[/itex] such that for all [itex]x \in U[/itex], if [itex]|x - a| < \delta[/itex] then [itex]|f(x) - f(a)| < \epsilon[/itex]. The definition does not require that [itex](a - \delta, a + \delta) \subset U[/itex]. Indeed it may happen that [itex]U \cap (a - \delta, a + \delta) = \{a\}[/itex], in which case [itex]f[/itex] being continuous as [itex]a[/itex] tells you nothing about its values at any other points in [itex]U[/itex]. Thus, for example, any [itex]f: U = (-\infty, 1] \cup \{2\} \cup [3, \infty) \to \mathbb{R}[/itex] is continuous at 2, because for all [itex]x \in U[/itex], if [itex]|x - 2| < \frac12[/itex] then [itex]x = 2[/itex]. Thus for all [itex]x \in U[/itex], if [itex]|x - 2| < \frac12[/itex] then [itex]|f(x) - f(2)| = 0[/itex]. [/QUOTE]
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Question regarding the continuity of functions
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