1. Let's start by stating the problem: We want to understand what a continuous differentiable function is. 2. A function $f(x)$ is called \textbf{continuous} at a point $x=a$ if the limit of $f(x)$ as $x$ approaches $a$ equals the function value at $a$, that is: $$\lim_{x \to a} f(x) = f(a)$$ This means there are no breaks, jumps, or holes in the graph of the function at $x=a$. 3. A function is \textbf{differentiable} at $x=a$ if the derivative $f'(a)$ exists. The derivative is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ This means the function has a well-defined tangent (slope) at $x=a$. 4. A function is \textbf{continuously differentiable} on an interval if it is differentiable at every point in that interval and its derivative function $f'(x)$ is continuous on that interval. 5. Important rule: If a function is continuously differentiable, it implies the function is smooth without sharp corners or cusps, and the slope changes smoothly. 6. In summary, a continuously differentiable function $f$ satisfies: - $f$ is continuous. - $f'$ exists everywhere in the domain. - $f'$ is continuous. 7. Example: The function $f(x) = x^2$ is continuously differentiable because: - It is continuous everywhere. - Its derivative $f'(x) = 2x$ exists everywhere. - The derivative $2x$ is continuous everywhere. This ensures $f(x) = x^2$ is smooth and has no breaks or sharp points. Final answer: A continuously differentiable function is one that is smooth, has no breaks, and whose derivative is also continuous, ensuring smooth changes in slope.