How to Find Discontinuity
To find and prove a discontinuity, it is useful to start with the definition of continuity. Continuity of f(x) at x0 means that, as x converges on x0, all f(x) converge on f(x0). Therefore, for a discontinuity, no matter how close x gets to x0, there's some x in between where f(x) is at least constant c away from f(x0). The general strategy is to guess and test an x0 where f(x) may be discontinuous. Then find neighboring f(x) which won't get closer than c.
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Instructions
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Finding and Proving and Discontinuity in a Function
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Guess where a discontinuity exists.
For example, if f(x)=0 for x<0 and 1 for non-negative x, an obvious guess as to where there's a discontinuity is at 0.
As a less trivial example, suppose f(x)=0 for rational numbers and x for irrational numbers. One would guess that any x other than zero has a discontinuity, since any continuous interval, no matter how small, has both rational and irrational numbers.
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Find a sequence of x's that converges to x0 without f(x) converging to f(x0).
For example, in the first example, any sequence of x's converging from the left side of x=0 will give a function value f(x)=0, while f(0)=1.
An example of such a sequence is x=-1/10, -1/100, -1/1000, ....
As for the nontrivial example, pick an x0 other than 0. Let's just keep calling it x0, so as not to lose generality. If x0 is irrational, then f(x0) is x0. Then a sequence of x's can be created which converge on x0, but for which f(x) doesn't converge on x0. This is easily done by making the x's all rational approximations of x0. For example, if x is the irrational number pi, the sequence of x's converging on x0 is 3, 3.1, 3.14, 3.141, 3.1415, .... The value of f(x) for all these x's is 0. No matter how close these x's get to x0=pi, f(x) won't converge to f(x0)=f(pi)=pi.
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3
Determine c, the minimum distance from f(x0) that some function value in the interval between x and x0 will have as x approaches x0.
In the first example c=1 since the f(x) were always 1 away from f(0)=0. In the second example, the sequence produced function values of 0 while f(x0)=f(pi)=pi. So c=pi.
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