Continuity discussion

[User16]

f(x+y)= (x+y)! , f being a function on the domain R where R consists of all the lines x+y=constant=k (say)
Can anyone discuss the continuity of the function f on the domain R in details ?

 

[Ishuda]

f(x+y)= (x+y)! , f being a function on the domain R where R consists of all the lines x+y=constant=k (say)
Can anyone discuss the continuity of the function f on the domain R in details ?

If we have x+y=k (a constant), then you just have the function at one point, but, as possibly you meant, x+y=k is the family of curves with parameter (variable) k then we still have
f(k) = k!
but now k varies and we have a function of k. The familiar factorial function defined on the non-negative integers can be extended, see
http://en.wikipedia.org/wiki/Gamma_function
for example:

The gamma function is defined for all complex numbers except the negative integers and zero. For complex numbers with a positive real part, it is defined via a convergent improper integral:

 

[User16]

If we have x+y=k (a constant), then you just have the function at one point, but, as possibly you meant, x+y=k is the family of curves with parameter (variable) k then we still have
f(k) = k!
but now k varies and we have a function of k. The familiar factorial function defined on the non-negative integers can be extended, see
http://en.wikipedia.org/wiki/Gamma_function
for example:

Thank you for the help