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f-x-x-sin-1-x-if-0-lt-x-1-0-if-x-0-Show-that-f-is-continous-but-not-of-bounded-variation-




Question Number 152362 by Tawa11 last updated on 27/Aug/21
f(x)   =    { ((x sin (1/x) ,          if     0   <   x   ≤   1)),((      0    ,                if   x   =   0)) :}  Show that   f   is continous but not of bounded variation
$$\mathrm{f}\left(\mathrm{x}\right)\:\:\:=\:\:\:\begin{cases}{\mathrm{x}\:\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{x}}\:,\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\:\:\:\:\mathrm{0}\:\:\:<\:\:\:\mathrm{x}\:\:\:\leqslant\:\:\:\mathrm{1}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\:\:\mathrm{x}\:\:\:=\:\:\:\mathrm{0}}\end{cases} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\mathrm{f}\:\:\:\mathrm{is}\:\mathrm{continous}\:\mathrm{but}\:\mathrm{not}\:\mathrm{of}\:\mathrm{bounded}\:\mathrm{variation} \\ $$

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