Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2020 Q.27)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Continuity of Functions - A piecewise function $f(x)$ is continuous at $x = x_0$ if and only if: $\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = f(x_0)$ **2. Detailed Step-by-Step Calculation**: - **Given Piecewise Function**: $f(x) = \begin{cases} c x & \text{if } x \ge 1 \\ 15 & \text{if } x < 1 \end{cases}$ - **Left-hand Limit ($x \to 1^-$)**: $\lim_{x \to 1^-} f(x) = 15$ - **Right-hand Limit ($x \to 1^+$)**: $\lim_{x \to 1^+} f(x) = c(1) = c$ - **Equating for Continuity at $x = 1$**: $c = 15$ **3. Final Answer**: - **Calculated Value**: **15** - **Official GATE Accepted Range**: **15 to 15**