In this paper, we introduce a class of functions that assume only a limited number $λ$ of values within a given Hamming $ρ$-ball and call them locally $(ρ, λ)$-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when $λ= 4$. We also demonstrate that any function can be represented as a locally $(ρ, λ)$-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.