## Abstract

We have been assuming up to now that the functions we were working with were “continuous.” That is, we have assumed that if ƒ(

*r*)=0 and*x*_{0},*x*_{1},*x*_{2}, … were numbers that got closer and closer to*r*, then the numbers ƒ(*x*_{0}), ƒ(*x*_{1}), ƒ(*x*2), … would get closer and closer to ƒ(*r*) = 0. More generally, a function ƒ is*continuous*if for each point*y*and each sequence*x*_{0},*x*_{1},*x*_{2}… =*x*_{2}that has*y*as limit,*x*_{2}→*y*, we have ƒ(*x*_{2})→ƒ(*y*). We may also express this by writing$$\mathop {\lim }\limits_{x \to y} f(x)\; = \;f(y)$$

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