MATH 401 - NOTES Sequences of functions Pointwise and ...

SPRING 2009 MATH 401 - NOTESS equences of functionsPointwise and Uniform ConvergencePreviously, we have studied Sequences ofreal numbers. Now we discusssequences of real-valued functions . By a sequence{fn}of real-valued func-tions onD, we mean a sequence (f1, f2, .. , fn, ..) such that eachfnis afunction having domainDand range a subset Pointwise a subset ofRand let{fn}be a sequence of functionsdefined onD. We say that{fn}converges Pointwise onDiflimn fn(x)exists for each other words, limn fn(x) must be a real number that depends only this case, we writef(x) = limn fn(x)for everyxinDandfis called thepointwise limit of the sequence{fn}.Formal Definition:The sequence{fn}converges Pointwise tofonDif foreveryx Dand for every >0, there exists a natural numberN=N(x, )such that|fn(x) f(x)|< whenevern > :The notationN=N(x, ) means that the natural numberNdependson the choice ofxand.

Hence {fn} is not uniformly convergent. Theorem. Let D be a subset of R and let {fn} be a sequence of continuous functions on D which converges uniformly to f on D. Then its limit f is continuous on D. Example 10. Let {fn} be the sequence of functions defined by fn(x) = cosn(x) for −π/2 ≤ x ≤ π/2. Discuss the uniform convergence of the ...

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