Limits, Continuity & Differentiability
Functional Equations and Derivatives
Grade 12
Question:
<p><strong>Ex. 10</strong> Let $f$ be a function such that $f(x + f(y)) = f(x) + y$, $\forall x, y \in \mathbb{R}$. Then find $f(0)$.</p><p>If it is given that there exists a positive real $h$, such that $f(h) = h$ for $0 \leq h \leq H$, then find $f'(x)$.</p>
<p>(a) $0, 1$</p>
<p>(b) $+1, 0$</p>
<p>(c) $2, 1$</p>
<p>(d) $+2, 0$</p>
Step-by-Step Solution
Key Concept: Substitute specific values into the functional equation to find $f(0)$. The consistency requirement that $f$ maps $\mathbb{R}$ to $\mathbb{R}$ forces $f(0) = 0$.
<p><strong>Solution:</strong> Let $x = 0, y = 0$ in $f(x + f(y)) = f(x) + y$</p><p>$f(0 + f(0)) = f(0) + 0$</p><p>$f(f(0)) = f(0)$</p><p>Let $f(0) = c$. Then $f(c) = c$.</p><p>Setting $y = 0$: $f(x + c) = f(x)$, which means $f$ is periodic with period $c$.</p><p>For the function to satisfy the given functional equation consistently, we must have $c = 0$, so $f(0) = 0$.</p><p>From the condition $f(h) = h$ for $0 \leq h \leq H$ and differentiability, we get $f'(x) = 1$.</p>
Correct Answer: A