Question

Suppose that a function ƒ is continuous in a region $\Omega$. Prove that any two primitives of ƒ (if they exist) differ by a constant.

Answer 1

If f:Ω→Cf:Ω→C is holomorphic, where ΩΩ is a domain, then: f′=0⟹ff′=0⟹f is constant.

Outline of proof:

Write f=u+ivf=u+iv. That f′=0f′=0 gives:

∂u∂x=∂u∂y=∂v∂x=∂v∂y=0

∂u∂x=∂u∂y=∂v∂x=∂v∂y=0

Since ΩΩ is open and connected, the region of R2R2 over which uu an vv are defined, which is identified with ΩΩ through the identification of CC and R2R2, is open connected. Connectedness implies "stairway-wise connectedness", i.e. that any two points in the set can be joined by a "stairway" (a union of arranged segments each parallel to either the xx-axis or the yy-axis). Now prove that uu and vv are constant on any such stairway, hence constant on the whole set. Hint: to do the latter thing, use, for a fixed yy, the function g(t)=u(t,y)g(t)=u(t,y), and for a fixed xx, the function h(t)=u(x,t)h(t)=u(x,t) (do the same for vv).