Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $d_{\mathbb{R}}$ be the distance on $\mathbb{R}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv). Continuity of a map from a subset of $\mathbb{R}$ to $\mathbb{R}$ is meant for the map from $(\mathbb{R},d_{\mathbb{R}})$ to $(\mathbb{R},d_{\mathbb{R}})$.

(i) Let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval with endpoints $a$ and $b$, let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$, and let $y\in\mathbb{R}$ satisfy

$$f(a)\le y\le f(b)\qquad\text{or}\qquad f(b)\le y\le f(a).$$

Then there is $c\in[a,b]$ with $f(c)=y$.

(ii) Let $I$ be an interval and let $f:I\to\mathbb{R}$ be continuous on $I$. Then the image $f(I)$ is an interval.