You have to know the definition of "uniform convergence".īut the theorems from this chapter will not be on the final exam. \(f_n\to f\) pointwise, \(f_n'\to g\) uniformly on \(\) \(\implies\) \(f\) continuously differentiable over \(\) and \(f'=g\). \(f_n\) Riemann integrable over \(\) \(\implies\) \(f\) Riemann integrable over \(\) and \(\int_a^b f_n \to \int_a^b f\)ĭifferentiation: For continuously differentiable functions \(f_n\) on \(\) \(f_n\) uniformly continuous on \(\) \(\implies\) \(f\) uniformly continuous on \(\) \(f_n\) continuous \(\implies\) \(f\) continuous Main results: (will not be on final exam): We say in this case: " \(f_n(x)\) converges to \(f(x)\) uniformly for \(x\in\)".
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