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This is because of a rigorously defined procedure called "extending by continuity". The function is continuous, your issue is just an artifact of notation.
by krastanov 13y ago
This is because of a rigorously defined procedure called "extending by continuity". The function is continuous, your issue is just an artifact of notation.
- jay-anderson 13y agoCan you explain this more? From what I understand the original function has division by '0' at those specific values of 'x'. So the final reduced form has a different domain where it is valid.
- krastanov 13y agoI will give an example, not the general definition. Consider the function f: x -> sin(x)/x. At x=0 you indeed have a division by 0, and if you stick to blindly using your notation it does not work. The function is not defined at x=0. However you usually (not always) are interested in the function as a whole. Look at the limit from the left, look at the limit from the right. They have the same values, it makes sense just to use this limiting value for the value at x=0. See https://en.wikipedia.org/wiki/Removable_singularity https://en.wikipedia.org/wiki/Removable_singularity