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Then Spivak says, f is uniformly continuous on [a,c]...

For prving this, he invokes the continuity of f on b.....

My questions here are:

1.For a given ε, we have a δ1 which works on whole of interval [a,b] and similarly, for this ε, δ2 works for all of [b,c].....Why can't we take δ=min(δ1,δ2) on this whole inteval [a,c], and consider that this δ will work for a given ε. Why does he consider continuity at b?

2. For this point b again, he considers these cases:

a. Either x<b<y or y<b<a....

b. ιx-bι<δ3 for ε/2 and ιy-bι<δ3 for ε/2

c. he takes ιx-yι< min(δ1,δ2,δ3), then he proves ιf(x)-f(y)ι<ε...

Why should he consider the following assumption that x and y only lie in (b-δ3,b+δ3)? What if for all x and y lying outside?