# Interactive Real Analysis

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## 6.3. Discontinuous Functions

### Examples 6.3.2:

Which of the following functions, without proof, has a 'fake' discontinuity, a 'regular' discontinuity, or a 'difficult' discontinuity ?
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This function seems to have a 'fake' discontinuity at x = 3, since we could easily move the single point at x = 3 to the right height, thereby filling in the discontinuity.
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This function, while simple, seems to have a 'true' discontinuity at x = 0. We can not change the function at a single point to make it continuous.
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This function is unclear. It is hard to determine what exactly is going on as x gets closer to zero. Assuming that the function does turn out to be discontinuous at x = 0, it definitely seems to have a 'difficult' discontinuity at x = 0.

This function is impossible to graph. The picture above is only a poor representation of the true graph. Nonetheless, given any point x, the function jumps between 1 and 0 in every neighborhood of x. That seems to mean that the function has a difficult discontinuity at every point.
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