Discontinuity Limits: Does f(b) Mean Discontinuous at x=b?

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SUMMARY

The discussion centers on the concept of removable discontinuities in functions, specifically addressing whether f(b) existing implies discontinuity at x=b. It establishes that if a function is continuous at a, then any sequence x_n converging to a must satisfy lim f(x_n) = f(a) as n approaches infinity. The presence of two paths converging to different limits demonstrates discontinuity at that point. Therefore, the existence of f(b) does not guarantee continuity at x=b.

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Loppyfoot
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On the graph, the limit at a has a removable discontinuity at b. And below this, there is a darkened circle, which means that f(b) exists. Does this mean that f is discontinuous at x=b?
 
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Use the fact that if a function is continuous at a, then any sequence x_n that converges to a will satisfy lim f(x_n) = f(a) as n tends to infinite. So if you can find two such paths that tend to different limits you will have shown that the function is discontinuous at that point
 

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