I am confused with the definition of modular functions. In some textbooks such as JP Serre's "Course in Arithmetic", a modular function of weight $2k$, is defined as a meromorphic function on the upper half plane of $\mathbb{C}$, s.t. it satisfies the modularity condition
$$f(z+1)=f(z)$$ and $$f(-1/z)=z^{2k}f(z)$$ and also meromorphic at infinity. Additionally if it is holomorphic at infinity then it is a modular form.
In some other books it is written as any modular form of weight zero is a modular function. They have defined a modular form of weight $2k$ to be any function acting on upper half of $\mathbb{C}$ that satisfies the modularity condition as above. So, basically a modular function is a function which satisfies
$$f(z+1)=f(z)$$ and $$f(-1/z)=f(z)$$ No extra meromorphicity or holomorphicity conditions were given.
How these two definition are the same?