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


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.


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