Removable Discontinuities At a removable discontinuity, the left-hand and right-hand limits are equal but either the function is not deﬁned or not equal to these limits: lim f (x) = lim− f (x) = � f (x0 ) x→x+ 0 x→x0 Figure 1: A removable discontinuity: the function is continuous everywhere except one point sin(x) x For example, g(x) = and h(x) = 1−cos are deﬁned for x �= 0, but x x both functions have removable discontinuities. This is not obvious at all, but we will learn later that: sin x 1 − cos x = 1 and lim = 0. x→ 0 x x→ 0 x lim So both of these functions have removable discontinuities at x = 0 despite the fact that the fractions deﬁning them have a denominator of 0 when x = 0.

Removable Discontinuities

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