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Physics LibreTexts

1.7: Continuity

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Continuity

Continuity is an important concept in the theory of real functions. A continuous function is one whose output f(x) does not undergo abrupt jumps when x changes by tiny amounts. A function can be continuous over its entire domain, or only a subset of its domain. For example, sin(x) is continuous for all x, whereas f(x)=1/x is discontinuous at x=0. Another function that is discontinuous at x=0 is the step function Θ(x)={1,forx00,otherwise. Mathematicians have even come up with functions that are discontinuous everywhere in their domain, but we won’t be dealing with such cases.

The rigorous definition of continuity is as follows:

Definition: Word

A function f is continuous at a point x0 if, for any ϵ>0, we can find a δ>0 such that setting x closer to x0 than a distance of δ brings f(x) closer to f(x0) than the specified distance ϵ.

That’s a very complicated sentence, and it may be easier to understand using this illustration:

clipboard_e1a9c8164cd862a5319ca0a2d198d2e75.png
Figure 1.7.1

A counter-example, with a function that has a discontinuity at some x0, is shown below:

clipboard_ea5bbf237edb95344df53506bf787813b.png
Figure 1.7.2

If we choose ϵ smaller than the gap, then no matter what value of δ>0 we try, any choice of 0<x<δ will give a value of f(x) that’s further than ϵ from f(x0). Hence, the continuity condition is violated for sufficiently small choices of ϵ=1/2, and we say that f is discontinuous at x0.


This page titled 1.7: Continuity is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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