Suppose f(x)=⌊x⌋ for x≥0. Define a sequence of functions (fn(x))n≥1 where
fn(x)={xn:x∈[0,1)(x−1)n+1:x∈[1,2)(x−2)n+2:x∈[2,3)⋮
Questions:
1) Is fn(x) continuous for all x≥0?
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 (x−k)n+k→k for x∈[k,k+1), since 0≤(x−k)<1.
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