My question is regarding differentiability of a scalar function of two variables.
Many calculus textbooks define differentiability at a point by requiring that the function be defined on an open interval containing that point, and many do not have this requirement. This got me thinking, can a function be differentiable on its boundary?
I think that functions with natural domain restrictions {Example: f(x,y) = sqrt( x - (y^2) ) } are not differentiable on their boarders. However, if I take a function like f(x,y) = x + y and restrict its domain to D = {(x,y): x >= 0 and y >= 0}, It is differentiable on its boarders.
My question is, is my thought process correct? If so, wouldn’t it be the case that the most general definition would not require f to be defined on an open interval containing the point?
Many calculus textbooks define differentiability at a point by requiring that the function be defined on an open interval containing that point, and many do not have this requirement. This got me thinking, can a function be differentiable on its boundary?
I think that functions with natural domain restrictions {Example: f(x,y) = sqrt( x - (y^2) ) } are not differentiable on their boarders. However, if I take a function like f(x,y) = x + y and restrict its domain to D = {(x,y): x >= 0 and y >= 0}, It is differentiable on its boarders.
My question is, is my thought process correct? If so, wouldn’t it be the case that the most general definition would not require f to be defined on an open interval containing the point?