A sequence of uniformly continuous functions that converge to a non-uniformly continuous function (CSIR) (INTERVIEW)

Give an example of a sequence of uniformly continuous functions that converges to a non-uniformly continuous function,

Solution:

Case 1: The limit function is not even continuous

Consider $f_n(x) = x^n$ on $[0,1]$ then $f_n$ converge pointwise to $f$ where $f(x) = \begin{cases} 0 \text{ if } x \in [0,1) \\ 1 \text{ if } x = 1\end{cases}$. Now, each $f_n$ is a continuous function on complact set and hence uniformly continuous but its pointwise limit is not even continuous.

Case 2: The limit function is continuous but not uniformly continuous

Consider the function $f(x) = x^2$ on $[0,\infty)$. We know that $x^2$ is uniformly continuous only on finite intervals and hence $f$ is not uniformly continuous on $[0,\infty)$. Define $f_n(x) = \begin{cases} f(x) \text{ if } 0 \le x \le n \\ f(n) \text{ if } x > n \end{cases}$, then clearly each $f_n$ is uniformly continuous and converges pointwise to the function $f$. This complete the proof.

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