Monday, 3 February 2014

real analysis - Construct a sequence of continuous functions which converges pointwise to lfloorxrfloor




Suppose f(x)=x for x0. Define a sequence of functions (fn(x))n1 where



fn(x)={xn:x[0,1)(x1)n+1:x[1,2)(x2)n+2:x[2,3)



Questions:



1) Is fn(x) continuous for all x0?



2) Does the function fn(x) converge pointwise to x?



If yes to both questions above, can we write fn(x) in a single function instead of piece-wise function?




My guess: Yes to both questions. But I am unable to express fn(x) in a single function.


Answer



Note that (xk)n+kk for x[k,k+1), since 0(xk)<1.


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