Paper 1, Section II, E

(a) Let $f:[a, b] \rightarrow \mathbb{R}$. Suppose that for every sequence $\left(x_{n}\right)$ in $[a, b]$ with limit $y \in[a, b]$, the sequence $\left(f\left(x_{n}\right)\right)$ converges to $f(y)$. Show that $f$ is continuous at $y$.

(b) State the Intermediate Value Theorem.

Let $f:[a, b] \rightarrow \mathbb{R}$ be a function with $f(a)=c<f(b)=d$. We say $f$ is injective if for all $x, y \in[a, b]$ with $x \neq y$, we have $f(x) \neq f(y)$. We say $f$ is strictly increasing if for all $x, y$ with $x<y$, we have $f(x)<f(y)$.

(i) Suppose $f$ is strictly increasing. Show that it is injective, and that if $f(x)<f(y)$ then $x<y .$

(ii) Suppose $f$ is continuous and injective. Show that if $a<x<b$ then $c<f(x)<d$. Deduce that $f$ is strictly increasing.

(iii) Suppose $f$ is strictly increasing, and that for every $y \in[c, d]$ there exists $x \in[a, b]$ with $f(x)=y$. Show that $f$ is continuous at $b$. Deduce that $f$ is continuous on $[a, b]$.

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