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f doesn't even have either of a right or left limit at 0. f is maybe (piece-wise) continuous over what pseudo-domain? R or R\{0}? You could axiomitize that an

by gowld 6y ago

f doesn't even have either of a right or left limit at 0. f is maybe (piece-wise) continuous over what pseudo-domain? R or R\{0}? You could axiomitize that an infinite discontinuity is like an unbouned function as x->infinity, but then how would you avoid 1/x being regular continuous? I think you are claimokg that a set being incomplete "at the end" is different from a set having a hole in the middle -- that the question of continuity presumes connected sets. That's not standard but might be an appropriate assumption for the context of your paper. Anyway, arguing over terminology is boring unless it raises conceptual issues. The point is to communicate, which has at least 2 stakeholders. Clarify your terms and move on.

Ericson2314 6y ago
The idea is that what 1/x is is intuitively simple enough, so we should have some standard terminology for it. And saying left and right limits is -infinity or +infinity really also isn't that weird. I'm pretty sure other metric spaces can be likewise extended and end up with similar algebraic laws as the "extended real numbers". Again this isn't very profound, but is good for education and efficient communication, and so should be perused. Finally, it's interesting that measure spaces with infinite measure is already a thing that people. I would like to see more connections with metric and measure spaces; e.g. we can have an n-point metric which is defined using the measure of the (n-1)-simplex. Just as regular metric spaces have a "triangle inequality", 3-point metric spaces should have a "tetrahedron inequality", and n-point metric spaces should have a "(n-1)-simplex inequality". Again, this is not profound, but good for communication, and connections between definitions help one compress everything for better mental storage.
tgb 6y ago
I think you missed the point of the example, which is that people rarely clarify this because to the author their definition seems obviously correct. I made it 90% through a PhD without having considered that there could be more than one possible meaning for "piece-wise continuous". The domain is R in this case. The less-restrictive definition would be that f is piece-wise continuous if there are a discrete set of points .. < x_0 < x_1 < ... with f continuous on each interval (x_i, x_{i+1}). The alternative definition is that, plus requiring that the restriction of f to those intervals have limits at the endpoint. For piecewise smooth function, there's an even larger variety of possible meanings, yet it's often stated without clarification.
- thaumasiotes 6y ago
  > The less-restrictive definition would be that f is piece-wise continuous if there are a discrete set of points .. < x_0 < x_1 < ... with f continuous on each interval (x_i, x_{i+1}). The alternative definition is that, plus requiring that the restriction of f to those intervals have limits at the endpoint. This sounds kind of surreal; putting them in shorter terms, we have two rival definitions for "piecewise continuous": 1. A function f is piecewise continuous if there exist one or more intervals over which f is continuous. 2. A function f is piecewise continuous if there exist one or more intervals over which f is (1) continuous, and (2) bounded. I agree that it sounds obvious which of those is more appropriate as a definition of "piecewise continuous"... (It also worries me that the definition you give requires the intervals to be adjacent, but doesn't require that more than one interval exist. A function that is continuous over (-2, -1) and also over (1, 2), but not anywhere else, would meet this definition, but you wouldn't be able to include both of those intervals of continuity in the set of endpoints. I would prefer to either have two sets of endpoints, such that we end up saying f is continuous over (x_i, y_i), (x_{i+1}, y_{i+1}), etc. (if we want to allow for intervals of discontinuity), or to say that the intervals (-inf, x_0) and (x_n, +inf) also count (if we don't). However, if we take that second approach, and we go with the definition of piecewise continuity that requires the function be bounded over every interval, we've just defined functions like f(x) = x as being not piecewise continuous despite the fact that they are continuous.)
  - tgb 6y ago
    I would generally want the intervals to cover the entire domain (or rather for their closures to cover it). And half-infinite (or all of R) intervals would also be allowed, but I couldn't think of a good way to state that concisely. And I would only require the limits at the finite end points so that boundedness is not a concern (piecewise continuity should be a local property anyway). Surprisingly complicated to specify fully!

by gowld 6y ago

Ericson2314 6y ago

The idea is that what 1/x is is intuitively simple enough, so we should have some standard terminology for it. And saying left and right limits is -infinity or +infinity really also isn't that weird. I'm pretty sure other metric spaces can be likewise extended and end up with similar algebraic laws as the "extended real numbers". Again this isn't very profound, but is good for education and efficient communication, and so should be perused. Finally, it's interesting that measure spaces with infinite measure is already a thing that people. I would like to see more connections with metric and measure spaces; e.g. we can have an n-point metric which is defined using the measure of the (n-1)-simplex. Just as regular metric spaces have a "triangle inequality", 3-point metric spaces should have a "tetrahedron inequality", and n-point metric spaces should have a "(n-1)-simplex inequality". Again, this is not profound, but good for communication, and connections between definitions help one compress everything for better mental storage.

tgb 6y ago

I think you missed the point of the example, which is that people rarely clarify this because to the author their definition seems obviously correct. I made it 90% through a PhD without having considered that there could be more than one possible meaning for "piece-wise continuous". The domain is R in this case. The less-restrictive definition would be that f is piece-wise continuous if there are a discrete set of points .. < x_0 < x_1 < ... with f continuous on each interval (x_i, x_{i+1}). The alternative definition is that, plus requiring that the restriction of f to those intervals have limits at the endpoint. For piecewise smooth function, there's an even larger variety of possible meanings, yet it's often stated without clarification.

thaumasiotes 6y ago
> The less-restrictive definition would be that f is piece-wise continuous if there are a discrete set of points .. < x_0 < x_1 < ... with f continuous on each interval (x_i, x_{i+1}). The alternative definition is that, plus requiring that the restriction of f to those intervals have limits at the endpoint. This sounds kind of surreal; putting them in shorter terms, we have two rival definitions for "piecewise continuous": 1. A function f is piecewise continuous if there exist one or more intervals over which f is continuous. 2. A function f is piecewise continuous if there exist one or more intervals over which f is (1) continuous, and (2) bounded. I agree that it sounds obvious which of those is more appropriate as a definition of "piecewise continuous"... (It also worries me that the definition you give requires the intervals to be adjacent, but doesn't require that more than one interval exist. A function that is continuous over (-2, -1) and also over (1, 2), but not anywhere else, would meet this definition, but you wouldn't be able to include both of those intervals of continuity in the set of endpoints. I would prefer to either have two sets of endpoints, such that we end up saying f is continuous over (x_i, y_i), (x_{i+1}, y_{i+1}), etc. (if we want to allow for intervals of discontinuity), or to say that the intervals (-inf, x_0) and (x_n, +inf) also count (if we don't). However, if we take that second approach, and we go with the definition of piecewise continuity that requires the function be bounded over every interval, we've just defined functions like f(x) = x as being not piecewise continuous despite the fact that they are continuous.)
- tgb 6y ago
  I would generally want the intervals to cover the entire domain (or rather for their closures to cover it). And half-infinite (or all of R) intervals would also be allowed, but I couldn't think of a good way to state that concisely. And I would only require the limits at the finite end points so that boundedness is not a concern (piecewise continuity should be a local property anyway). Surprisingly complicated to specify fully!