What is the function that is either invariant or shifted if its input is the output of any function varying in input scale?

Question

I would like to know if there is any function f which satisfies any of two conditions (for any g) described below:

1. f( g(x) ) = f( g(cx) )

2. f( g(x) ) = f( g(cx) + h(c) )

*h can be any function since it just describes that the scaling causes the shift.

I was thinking about f that convolutes g with normalized Laplacian of Gaussian, but I guess it is unsatisfactory. Thank you ahead.