On some papers you read online you will find theorems dabbled with:
"Under a relatively mild condition, ...."
What do mathematicians mean by this and what are some examples of correct usage?
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2$\begingroup$ Some theorems require extremely stringent conditions. I'm not sure about your background so I'm not sure about an example, but "mild" conditions usually suggest that most of the "day to day" objects that the subject is dealing with satisfy the conditions for a particular theorem. $\endgroup$– Andres MejiaFeb 15, 2016 at 2:47
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6$\begingroup$ A "relatively mild condition" is a condition that is usually satisfied in practice. So you don't need to worry that this condition is going to prevent you from using whatever result is being stated. $\endgroup$– littleOFeb 15, 2016 at 2:47
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$\begingroup$ the hausdorff condition on a topological space is usually considered to be a mild extra condition $\endgroup$– Andres MejiaFeb 15, 2016 at 2:49
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$\begingroup$ An example might be an assumption that a function is continuous or differentiable versus, say, smooth (infinitely differentiable) or analytic. Usually counter-examples will be a bit pathological. $\endgroup$– Derek Elkins left SEFeb 15, 2016 at 2:49
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1$\begingroup$ @Stefan Agreed, context (and subculture) is everything. $\endgroup$– BrianOFeb 15, 2016 at 5:01
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1 Answer
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A condition that is not too hard to get, or one that is in practice always fulfilled.
For example in many circumstances a function being bounded or measurable or continuous.
A mild condition would be the opposite of a strong condition.
Being continuous is a more mild condition than being analytic, which is a stronger condition than being smooth.