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Ankit Rana (936)

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 given that  f(x) . f[f(x] = 1

and f(1000) = 999

hence for x = 1000 we have f(1000) . f [ f(1000) ] = 1 

i.e. f(1000) f(999) = 1

hence f (999) = 1/f(1000) = 1/999

now since f is a continous function

it must attain all values between 1/999 and 999 in the interval [999,1000] since f(999) = 1/999 and f(1000) = 999

hence for some x between 999 and 1000 the function will attain 500 such that f(x) = 500

hence we have for that x    f(x) . f[ f(x)] = 500 f(500) = 1

hence f(500) = 1/500

here u need not know the function but u can definitely say that f(500) = 1/500 since we are using the property of continous functions.
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