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Let R be the set of real numbers and f :...

Let R be the set of real numbers and `f : R to R` be such that for all x and y in R, `f(x) -f(y)|^(2) le (x-y)^(3)`. Prove that f(x) is a constant.

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Since, `|f(y) - f(x) |^(2) le (x - y)^(3) `
`rArr" "(|f(y)-f(x)|^(2))/((y-x)^(2) le (x-y)`
` rArr" "|(f(y)-f(x))/(y-x)|^(2) le x - y` …(i)
` rArr" " underset( y to x) lim |(f(y) - f(x))/(y-x)|^(2) le underset( y to x) lim (x - y) `
` rArr" " |f'(x) |^(2) le 0`
which is only possible, if `|f'(x)| = 0`
`:. f'(x) = 0`
or f'(x) = Constant
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