I understand you question as,Originally Posted by rachael
is a real-valued function. It is definitely not countinous on because 5 is not in its domain. Thus, you are asking how do define the function at thus it would be countinous? Notice you function is, = . This a linear function not defined at . Thus, we need to redefine it at call that function . We need not do redefine it anywhere else because it is linear and is countinous. By the definition of countinuity we need that,
Since, and are function that agree at all but one point (at 5) we have that they have the same limit. Thus, , but . Thus, we have . Thus, the function,
is the redefined countinous function of .