Continuous function and definition

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A function described as continuous typically implies it is defined over the real numbers unless otherwise specified. Continuity means the function's value matches the limit at that point. However, there can be implicit restrictions on the domain that are not stated, such as in the case of f(x) = 1/x, which is only defined for nonzero reals. Therefore, while a continuous function is generally assumed to be defined over R, one must consider potential domain restrictions. Understanding these nuances is crucial for accurately interpreting function definitions in mathematics.
gummz
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If a function is continuous (nothing else specified), is it defined over R? Continuity means a function's value being the same as the limit for that point IIRC, but I don't know if it being continuous (over R presumably) means that it is also defined over R, or just that it's continuous wherever it is defined.
 
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gummz said:
If a function is continuous (nothing else specified), is it defined over R? Continuity means a function's value being the same as the limit for that point IIRC, but I don't know if it being continuous (over R presumably) means that it is also defined over R, or just that it's continuous wherever it is defined.
If a function is defined as being continuous, with no restrictions, it's probably safe to assume that it is defined for all reals. Sometimes there can be one or more restrictions on the domain that aren't explicitly stated, such as for f(x) = 1/x, whose domain is all nonzero reals.
 

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