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Let f : R to R be a function such that ...

Let ` f : R to R` be a function such that
`f(x+y) = f(x)+f(y),Aax, y in R.` If f (x) is differentiable at x = 0, then

A

f(x) is differentiable in a finite interval containing zero

B

f(x) is continuous for all ` x in R`

C

f'(x) is constant for all ` x in R`

D

f(x) is differentiable except at finitely many points

Text Solution

Verified by Experts

The correct Answer is:
B, C
`f(x+y) = f(x)+ f(y)`, as f(x) is differentiable at x = 0 .
`rArr" "f'(0) = k` …..(i)
Now, `f'(x) = underset( h to 0) lim (f(x+h)-f(x))/h `
` underset( h to 0) lim (f(x)+f(h) - f(x))/h `
` = lim (h to 0) lim (f(h))/h" "[0/0" form"]`
Given, `f(x+y) = f(x) +f(y),AA x , y`
`:." " f(0) = f(0) + f(0)`,
when`" " x = y = 0 rArr f(0) = 0)`
Using L'Hospital's rule,
`= underset( h to 0) lim (f'(h))/1 = f'(0) = k` ....(ii)
`rArr` f'(x) = k, integrating both sides,
`f(x)= kx+C, "as " f(0) = 0`
`:.` f(x) is continuous for all ` x in R and f'(x) = k, ` i.e. constant for all ` x in R`.
Hence, (b) and (c ) are correct.
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