ANSWER
Let ϵ>0. By the definition, we need to show there exists δ>0 , such that
a≥5,b≥5 and ∣a−b∣<δ imply ∣f(a)−f(b)∣<ϵ (1)
The function f(x)=x21 is differentiable on the set [5,∞) . f′(x)=x3−2 and ∣f′(x)∣≤532<1 for all x∈[5,∞). Let b>a≥5 , according to the mean value theorem, there exists point c∈(a,b) such that f(b)−f(a)=f′(c)⋅(b−a). So, for all a,b∈[5,∞) we have
∣f(b)−f(a)∣≤∣b−a∣ .
Hence, if ϵ>0 and δ=ϵ , then
∣a−b∣<δ=ϵ⇒∣f(b)−f(a)∣≤∣b−a∣<δ=ϵ
Those, (1) is satisfied, which means that the function is uniformly continuous.
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