1. The problem asks to sketch a function $f$ with specific limit and continuity properties: - $\lim_{x \to 2^-} f(x) = 1$ and $\lim_{x \to 2^+} f(x) = 3$ (a jump discontinuity at $x=2$). - $\lim_{x \to 3} f(x) = 2$ and $f$ has a removable discontinuity at $x=3$. - $f$ is continuous but not differentiable at $x=-1$. 2. Important concepts: - A jump discontinuity means the left and right limits at a point exist but are not equal. - A removable discontinuity means the limit exists at a point but the function value is either not defined or not equal to the limit. - Continuity at a point means the limit equals the function value. - Non-differentiability at a point can occur if the function has a sharp corner or cusp there. 3. To satisfy these: - At $x=2$, define $f(x)$ so that the left limit is 1 and the right limit is 3, e.g., $f(x) = 1$ for $x<2$ and $f(x) = 3$ for $x>2$. - At $x=3$, define $f(3)$ different from 2 or undefined, but ensure $\lim_{x \to 3} f(x) = 2$. - At $x=-1$, define $f$ continuous but with a sharp corner, e.g., $f(x) = |x+1|$ near $-1$. 4. Example piecewise function: $$ f(x) = \begin{cases} 1 & x < 2 \\ 3 & 2 < x < 3 \\ 0 & x = 3 \\ |x+1| & x \text{ near } -1 \\ \text{other values defined to make } f \text{ continuous elsewhere} \end{cases} $$ 5. This function meets all the conditions: jump at 2, removable discontinuity at 3, continuous but not differentiable at -1. Final answer: The function described above satisfies all the given properties.