1. **State the problem:** Determine which of the given functions are well-behaved. A function is considered well-behaved if it is continuous, differentiable, and does not have abrupt changes or discontinuities. 2. **Recall important rules:** - Continuity means no breaks or jumps in the function. - Differentiability means the function has a defined slope everywhere in its domain. - Piecewise functions must be checked at boundary points for continuity and differentiability. 3. **Analyze each function:** **A.** $u = x$ for $x \geq 0$, and $u = 0$ otherwise. - This is a piecewise function with a jump at $x=0$. - Left limit at 0 is 0, right limit at 0 is 0, so continuous at 0. - Derivative from left is 0, from right is 1, so not differentiable at 0. - **Not well-behaved due to non-differentiability at $x=0$.** **B.** $u = x^2$ - Polynomial, continuous and differentiable everywhere. - **Well-behaved.** **C.** $u = e^{-\lvert x \rvert}$ - Exponential of absolute value, continuous everywhere. - Differentiable everywhere except possibly at $x=0$. - Check derivative at 0: - Left derivative: $\frac{d}{dx} e^{-(-x)} = e^{-x}$ at 0 is $-1$. - Right derivative: $\frac{d}{dx} e^{-x} = -e^{-x}$ at 0 is $-1$. - Derivatives match, so differentiable at 0. - **Well-behaved.** **D.** $u = e^{-x}$ - Exponential function, continuous and differentiable everywhere. - **Well-behaved.** **E.** $u = \cos(x)$ - Trigonometric function, continuous and differentiable everywhere. - **Well-behaved.** **F.** $u = \sin(\lvert x \rvert)$ - Continuous everywhere. - Check differentiability at 0: - Left derivative: $\cos(\lvert x \rvert) \cdot \frac{d}{dx}(-x) = -\cos(0) = -1$. - Right derivative: $\cos(0) \cdot 1 = 1$. - Derivatives do not match at 0. - **Not well-behaved due to non-differentiability at $x=0$.** **G.** $u = \exp[-x^2]$ - Exponential of a polynomial, continuous and differentiable everywhere. - **Well-behaved.** **H.** $u = 1 - x^2$ for $-1 \leq x \leq 1$, and $u = 0$ otherwise. - Piecewise function with potential discontinuities at $x=\pm 1$. - Check continuity at $x=1$: - Left limit: $1 - 1^2 = 0$. - Right limit: $0$. - Continuous at 1. - Check differentiability at 1: - Left derivative: $-2x$ at 1 is $-2$. - Right derivative: derivative of constant 0 is 0. - Not differentiable at $x=1$. - Similarly at $x=-1$. - **Not well-behaved due to non-differentiability at $x=\pm 1$.** **Final summary:** - Well-behaved: B, C, D, E, G - Not well-behaved: A (non-differentiable at 0), F (non-differentiable at 0), H (non-differentiable at $\pm 1$)