Real Analysis : Jump Discontinuity
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Let f:R->R be increasing. Prove that if lim f(x) as x->c^+ and if lim f(x) as x->c^- must each exist at every point c belong to R. Argue that the only type of discontinuity a monotone function can have is a jump discontinuity.
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Proof:
For each c in R, we consider x_n=c-1/n, then x_n->c^- as n->oo, where oo denotes infinity. Since f is increasing, then f(x_n)<=f(x_(n+1)). But f(x_n)<=f(c) for each n. Thus f(x_n) is a bounded ...
Solution Summary
Jump Discontinuities are investigated. The solution is concise.
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