Exercise 4.5.3

Question

A function $f$ is increasing on $A$ if $f(x) \leq f(y)$ for all $x<y$ in $A$. Show that if $f$ is increasing on $[a, b]$ and satisfies the intermediate value property (Definition 4.5.3), then $f$ is continuous on $[a, b]$.

Ulisse Mini · Accepted Answer

Let $x \in [a,b]$ and choose $\epsilon > 0$.
  Let $L \in (f(a), f(x)) \cap (f(x)-\epsilon, f(x))$ IVP lets us find $c \in (a, x)$ with $f(c)= L$, thus $|f(x) - f(c)| < \epsilon$.
  Likewise we can find $d \in (x,b)$ with $|f(d)-f(x)|<\epsilon$.
  Because $f$ is increasing $f(x)-f(c) < \epsilon$ implies $f(x)-f(c')<\epsilon$ for $c' \in (c, x)$ (and likewise for $d$) meaning every $y \in (c, d)$ has $|f(x)-f(y)|<\epsilon$. To get a $\delta$-neighborhood simply set $\delta = \min\{x-c, d-x\}$.

Exercise 4.5.3

Answers

Comments

Exercise 4.5.3

Answers

Comments

Add answer