In order to be differentiable, we must satisfy two conditions. A function must be continuous and smooth. Please give an example of a function that does not satisfy continuity and a function that does not satisfy smoothness.
The Correct Answer and Explanation is:
Answer:
A function that does not satisfy continuity is:1 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases} \] A function that does not satisfy *smoothness* (but is still continuous) is: \[ f(x) = |x| \] — **Explanation (approx. 300 words):** Differentiability of a function at a point requires two essential conditions to be met: the function must be *continuous* at that point, and it must be *smooth*, meaning that there is no sharp turn or cusp in the graph. Both conditions are necessary but not sufficient on their own. First, consider continuity. A function is continuous at a point if the limit as the input approaches the point from both sides exists and is equal to the value of the function at that point. If a function is not continuous, it cannot be differentiable at that point. For example, the piecewise function \[ f(x) = \begin{cases} 1 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases} \] has a jump discontinuity at \( x = 0 \). As the function approaches zero from the left, the value is 1; from the right, the value is 2. The limit does not exist at that point, so the function is not continuous and thus not differentiable at \( x = 0 \). Next, consider smoothness. A function may be continuous at a point but still fail to be differentiable if there is a sharp corner or cusp. The function \( f(x) = |x| \) is continuous everywhere, including at \( x = 0 \), but it is not differentiable at that point. This is because the slope of the graph changes abruptly from negative on the left to positive on the right. The left-hand derivative at \( x = 0 \) is -1, and the right-hand derivative is 1. Since the left-hand and right-hand derivatives are not equal, the derivative does not exist at that point. Thus, both continuity and smoothness are required for differentiability. A failure in either condition prevents a function from being differentiable.
