Exercise 4.4.5

Question

Assume that $g$ is defined on an open interval $(a, c)$ and it is known to be uniformly continuous on $(a, b]$ and $[b, c)$, where $a<b<c$. Prove that $g$ is uniformly continuous on $(a, c)$.

Ulisse Mini · Accepted Answer

Let $\epsilon > 0$ and choose $\delta_1$ so that every $x,y \in (a,b]$ with $|x-y|<\delta_1$ has $|f(x)-f(y)|<\epsilon/2$, likewise choose $\delta_2$ so that every $x,y \in [b,c)$ with $|x-y|<\delta_2$ has $|f(x)-f(y)|<\epsilon/2$.
  Finally let $\delta = \min\{\delta_1, \delta_2\}$. The final case is if $x \in (a,b]$ and $y \in [b,c]$ where we use the triangle inequality
  $$
  |f(x) - f(y)| \le |f(x) - f(b)| + |f(b) - f(y)| < \epsilon/2 + \epsilon/2 = \epsilon
  $$
  Thus $f$ is uniformly continuous on $(a,c)$.

Exercise 4.4.5

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Exercise 4.4.5

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