A function defined as continuous without any restrictions is assumed to be defined over the entire set of real numbers (R). Continuity implies that the function's value at a point equals the limit of the function as it approaches that point. However, it is crucial to note that certain functions, like f(x) = 1/x, have implicit restrictions on their domains, which must be considered when discussing continuity.
PREREQUISITESStudents of calculus, mathematicians, and educators seeking to clarify the concepts of continuity and function domains.
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.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.