Restriction Stability of Continuity and of the Derivative

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure; for $s,t\in\mathbb{R}$ write $s<t$ to mean that $s\le t$ and $s\ne t$.

1. (Continuity) Let $B\subseteq A\subseteq X$, let $f:A\to Y$, let $x\in B$, and let $f|_{B}:B\to Y$ denote the restriction of $f$, given by $f|_{B}(y)=f(y)$ for $y\in B$. If $f$ is continuous at $x$ relative to $A$, then $f|_{B}$ is continuous at $x$ relative to $B$. Consequently, if $f$ is continuous on $A$, then $f|_{B}$ is continuous on $B$.

2. (Derivative) Let $I$ and $J$ be intervals with $J\subseteq I$, let $f:I\to\mathbb{R}$, and let $x_0\in J$ be an interior point of $J$. Then $x_0$ is an interior point of $I$; and if $f$ is differentiable at $x_0$, then the restriction $f|_{J}:J\to\mathbb{R}$ is differentiable at $x_0$ and

$$(f|_{J})'(x_0)=f'(x_0) .$$