I have a function $f$ defined as $f(x) = 1$ when $x = 0$ and $0$ elsewhere. There is a sequence of smooth functions $\{f_n\}$, such that $f_n\to f$ pointwise. Also $V(f_n)\to V(f) = 2$ where $V(f)$ is the total variation of function $f$ in the inetrval $(-1,1)$. Does the sequence of functions $\{f_n'\}$ converge? How do $f_n'$ look like when $n$ is large? Here $f_n'$ is the derivative of $f_n$.
A question related to convergence of a sequence of functions and their derivatives.
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real-analysis
1 Answers
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In a typical example you might construct, both $f_n$ and $f'_n$ will be near $0$ when $x$ is not near $0$, with $f$ having a large positive "spike" near $x=0$, and $f'$ correspondingly being large and positive in some interval slightly to the left of $0$, large and negative in some interval slightly to the right of $0$.
However, things can be somewhat more complicated, e.g. it's also not hard to find examples where $f'_n(x) \to \pm\infty$ as $n \to \infty$ for some infinite set of values of $x$.
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0Can we say $||f_n||_{L^{\infty}} \to 1$? – 2017-01-20
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0Yes we can. It's at least $1$ because $f_n(0) \to 1$. It's at most $1$ because otherwise $\limsup_n V(f_n) > 1$. – 2017-01-20
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0Let $S(f_n)$ = sum of absolute values of $f_n$ at its maxima and minima, can we say $S(f_n) \to 1$ – 2017-01-22
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0@ RobertIsrael : Related new question of mine http://math.stackexchange.com/q/2132785/2987 – 2017-02-07