Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $M$ be a subset of $X$, and let $T \colon X \to Y$ be a mapping that is uniformly continuous on $M$. Then is $T$ also continuous on $M$?
Definition of Continuity:
Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $T \colon X \to Y$ be a mapping, let $M$ be a subset of $X$, and let $p$ be a point of $X$. Then $T$ is said to be continuous at point $p$ if, for every real numnber $\varepsilon > 0$, we can find a real number $\delta > 0$ such that $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in X$ for which $$ d_X(x, p) < \delta.$$
If $T$ is continuous at each point $p \in M$, then $T$ is said to be continuous on set $M$.
Finally, if $T$ is continuous on $X$, then $T$ is said to be continuous.
Definition of Uniform Continuity:
Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $T \colon X \to Y$ be a mapping, and let $M$ be a subset of $X$. Then $T$ is said to be uniformly continuous on $M$ if, for every real number $\varepsilon > 0$, we can find a real number $\delta > 0$ such that $$d_Y\left( T(x_1), T(x_2) \right) < \varepsilon$$ for all points $x_1, x_2 \in X$ for which $$d_X\left( x_1, x_2 \right) < \delta.$$
Are these two sets of definitions correct?
Suppose $T$ is uniformly continuous on $M$, and let $p$ be an arbitrary point of $M$. So, for every real $\varepsilon > 0$, there exists $\delta > 0$ such that $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in M$ for which $$d_X\left( x, p \right) < \delta.$$
Thus the restriction $T \vert_M$ of the mapping $T$ to set $M$ is continuous at point $p$ and hence continuous on the set $M$.
How to show from here that $T$ is continuous at point $p$?
The answer is yes and there's nothing to show really: Note that the choice of $\delta$ in the definition of uniform continuity depends on $\epsilon$ only and hence satisfies the definition of pointwise continuity at any point at the same time.
In other words, in the definition of continuity $\delta$ is a function of both $p$ and $\epsilon$; uniform continuity is the special case, where $\delta$ can be chosen to be constant in $p$, hence is a function of $\epsilon$ only.