Analysis II | Part IB, 2005

Define uniform continuity for a real-valued function defined on an interval in R\mathbf{R}.

Is a uniformly continuous function on the interval (0,1)(0,1) necessarily bounded?

Is 1/x1 / x uniformly continuous on (0,1)(0,1) ?

Is sin(1/x)\sin (1 / x) uniformly continuous on (0,1)(0,1) ?

Justify your answers.

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