(a) Let $X \subset \mathbb{R}^{n}$ be a manifold and $p \in X$. Define the tangent space $T_{p} X$ and show that it is a vector subspace of $\mathbb{R}^{n}$, independent of local parametrization, of dimension equal to $\operatorname{dim} X$.

(b) Now show that $T_{p} X$ depends continuously on $p$ in the following sense: if $p_{i}$ is a sequence in $X$ such that $p_{i} \rightarrow p \in X$, and $w_{i} \in T_{p_{i}} X$ is a sequence such that $w_{i} \rightarrow w \in \mathbb{R}^{n}$, then $w \in T_{p} X$. If $\operatorname{dim} X>0$, show that all $w \in T_{p} X$ arise as such limits where $p_{i}$ is a sequence in $X \backslash p$.

(c) Consider the set $X_{a} \subset \mathbb{R}^{4}$ defined by $X_{a}=\left\{x_{1}^{2}+2 x_{2}^{2}=a^{2}\right\} \cap\left\{x_{3}=a x_{4}\right\}$, where $a \in \mathbb{R}$. Show that, for all $a \in \mathbb{R}$, the set $X_{a}$ is a smooth manifold. Compute its dimension.

(d) For $X_{a}$ as above, does $T_{p} X_{a}$ depend continuously on $p$ and $a$ for all $a \in \mathbb{R}$ ? In other words, let $a_{i} \in \mathbb{R}, p_{i} \in X_{a_{i}}$ be sequences with $a_{i} \rightarrow a \in \mathbb{R}, p_{i} \rightarrow p \in X_{a}$. Suppose that $w_{i} \in T_{p_{i}} X_{a_{i}}$ and $w_{i} \rightarrow w \in \mathbb{R}^{4}$. Is it necessarily the case that $w \in T_{p} X_{a}$ ? Justify your answer.