I made a proof for the following question below, but I am not sure how to complete it.
Let $z \in \mathbb{R}^7$ and $B=\mathbb{R}^7-\{z\}=\{x \in \mathbb{R}^7: x \not = z\}$. Suppose $g:B \to \mathbb{R}$ be a uniformly continuous function on $B$. Show that there exists a continuous function $g^*:\mathbb{R}^7 \to \mathbb{R}$ such that $g(x)=g^*(x)$ for all $x \in B$.
(note though not part of the problem explicitly I feel like I need to give some context to some readers. I know I need to use the concepts of Cauchy sequences of uniformly continuous functions, and sequential continuity are required to complete the proof).
My Proof:
Let $\{x_n\}$ be a Cauchy sequence of vectors in $B$, which converges to the vector $z \in \mathbb{R}^7$. Also let $\{g(x_n)\}$ be a sequence in $\mathbb{R}$. As $g$ is uniformly continuous $\{g(x_n)\}$ is a Cauchy sequence and as $\mathbb{R}$ is complete $\{g(x_n)\}$ converges to a point $g(z)=a \in \mathbb{R}$.
To show that $g^*$ is continuous for all $x \in B$, we take a second sequence $\{y_n\}$ of vectors in $\mathbb{R}^7$ which converges to a point $x$ in $\mathbb{R}^7$. I know I need to show here that $g^*$ is sequentially continuous at $x$ but I am not sure how to do this.