Let f : R \rightarrow_{}^{} R be a function such that f (x + y) = f (x) + f (y), \forall x, y \in R. If f(x) is differentiable at x = 0, then

Engineering

Mathematics

Derivative of Functions Expressed in The Determinant From

Question

Let f : R $\to_{}^{}$ R be a function such that

f (x + y) = f (x) + f (y), $\forall$ x, y $\in$ R.

If f(x) is differentiable at x = 0, then

f (x) is continuous $\forall x \in$ R.

f (x) is differentiable except at finitely many points.

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

f '(x) is constant $\forall x \in$ R.

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f(x + y) = f(x) + f(y)

By Partial differentiation with respect to x

f ' (x + y) = f ' (x)

f ' (y) = f '(0)

f(y) = (f '(0))y + c

f(y) = ky +c

$∴$ f(y) = ky as f(0) = 0

$∴$ f(x) = kx

Alternate

$f' (x) = \lim_{x \to \infty} \frac{f (x + h) - f (x)}{h}$

$= \lim_{x \to \infty} \frac{f (x) + f (h) - f (x)}{h} = \lim_{x \to \infty} \frac{f (h)}{h}$

= $λ$ (let)

f(x) = $λ$ x + c As f(0) = 0 ⇒c = 0

f(x) = $λ$ x

* The most appropriate answer to this question is (B, C), but because of ambiguity in language, IIT has declared (BC,BCD) as correct answer.