Step 1: Concept Overview:
The given inequality represents a Lipschitz condition, restricting the function's rate of change. This allows us to determine the function's derivative.
Step 2: Key Approach:
We utilize the derivative's definition:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
The provided condition is \( |f(a) - f(b)| \le (a-b)^2 \). We apply this to the derivative's definition.
Step 3: Detailed Reasoning:
Let \( y = x+h \). Then \( y-x = h \).The condition becomes \( |f(x+h) - f(x)| \le h^2 \).Now, consider the derivative's magnitude. For \( h eq 0 \):\[ \left| \frac{f(x+h) - f(x)}{h} \right| = \frac{|f(x+h) - f(x)|}{|h|} \]Using the given inequality:\[ \frac{|f(x+h) - f(x)|}{|h|} \le \frac{h^2}{|h|} = |h| \]Therefore, \( \left| \frac{f(x+h) - f(x)}{h} \right| \le |h| \).Taking the limit as \( h \to 0 \) to find the derivative:\[ |f'(x)| = \left| \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \right| = \lim_{h \to 0} \left| \frac{f(x+h) - f(x)}{h} \right| \]Using the squeeze theorem:\[ 0 \le \lim_{h \to 0} \left| \frac{f(x+h) - f(x)}{h} \right| \le \lim_{h \to 0} |h| = 0 \]This implies \( |f'(x)| = 0 \), meaning \( f'(x) = 0 \) for all \( x \in \mathbb{R} \).If the derivative is zero everywhere, the function is constant.\[ f(x) = C \]This eliminates options A, C, and D.
Step 4: Conclusion:
The function \(f(x)\) is a constant function.