Does it make sense to all functions or only those who are continuously derivable?
motoole2
Functions do not have to be differentiable or smooth. For example, I can define a function f(x) = {0 if x < 0.5, 1 if x >= 0.5} and g(x) = x, and we can still easily add or scale these functions.
However, we do have to be a little careful in practice, especially when dealing with functions that have nasty singularities. For example, the slide on the L2 norms of a function assumes that the square of a function has a well-defined integral, and this may not always the case. Suppose we define a function f(x) = x^{-1/2} over the open interval (0,1). The function is defined everywhere on this interval and it is differentiable, but its L2 norm \int_{x = 0 to 1} 1/x dx does not exist.
Does it make sense to all functions or only those who are continuously derivable?
Functions do not have to be differentiable or smooth. For example, I can define a function
f(x) = {0 if x < 0.5, 1 if x >= 0.5}andg(x) = x, and we can still easily add or scale these functions.However, we do have to be a little careful in practice, especially when dealing with functions that have nasty singularities. For example, the slide on the L2 norms of a function assumes that the square of a function has a well-defined integral, and this may not always the case. Suppose we define a function
f(x) = x^{-1/2}over the open interval(0,1). The function is defined everywhere on this interval and it is differentiable, but its L2 norm\int_{x = 0 to 1} 1/x dxdoes not exist.