# problems about complex-valued functions and continuity

I'm doing some homework proofs of continuity in complex-valued functions and need some help with 3 proofs.

1.-If $$f:U\subset\mathbb{C}_z\mapsto\mathbb{C}_{w_1}$$, $$g:W_1\subset\mathbb{C}_{w_1}\mapsto\mathbb{C}_{w_2}$$ are continuos functions, then, the composition $$g\circ f:U\mapsto\mathbb{C}_{w_2}$$ is also continuos. Here I have the hint that $$f(U)\subseteq W_1$$.

2.- A function $$f(z)=u(x,y)+iv(x,y)$$ is continuos in $$z_0=x_0+iy_0$$ iff it's real part $$u(x_0,y_0)$$ and it's imaginary part $$v(x_0,y_0)$$ are continuos in this point.

3.-If a function $$f(z)$$ is continuos in a bounded and closed set $$A$$, then it's uniformly continuos in $$A$$.

I was trying to aproach the first proof as the real-valued version of the theorem, but I'm not sure if it's ok, also don't know how to aproach the second part, and I think there's a similar real-valued version of part 3, but i'm not sure either

## 1 Answer

2/3 response

For 1. Since we have $$(g\circ f)^{-1}=f^{-1}\circ g^{-1}$$ for any map's composition, then for any $$O$$ an open set in $$\mathbb C_{w_2}$$ we have $$g^{-1}O$$ is open in $$\mathbb C_{w_2}$$ because $$g$$ is continuous, further, since $$f^{-1}(g^{-1}O)$$ is open in $$\mathbb C_z$$ because $$f$$ is continuous, then $$(g\circ f)^{-1}O$$ is open too, so $$g\circ f$$ is continuous.

For 2. You can see $$u=f\circ{\rm proj_1}$$ and $$v=f\circ{\rm proj_2}$$ and since both projections are continuous then $$u,v$$ are too.