The correct answer is: The function $y = |2 – 3x|$ is continuous for all real numbers $x$ and differentiable for all real numbers $x$ except $x = \frac{3}{2}$.
A function is continuous if it has no breaks or holes. A function is differentiable if it has a tangent line at every point.
The function $y = |2 – 3x|$ is continuous for all real numbers $x$ because it is defined for all real numbers $x$ and its graph has no breaks or holes.
The function $y = |2 – 3x|$ is differentiable for all real numbers $x$ except $x = \frac{3}{2}$ because the derivative of $y$ is $3 – 6x$, which is not defined at $x = \frac{3}{2}$.
Here is a graph of the function $y = |2 – 3x|$:
[asy]
unitsize(1 cm);
draw((-3,0)–(3,0));
draw((0,-2)–(0,2));
draw(graph(y=abs(2-3*x),-3,3),red);
label(“$y$”, (3,2), E);
label(“$x$”, (2,0), S);
label(“$y=|2-3x|$”, (1,1), N);
[/asy]
As you can see, the graph of the function is continuous for all real numbers $x$. However, the graph has a sharp turn at $x = \frac{3}{2}$. This means that the function is not differentiable at $x = \frac{3}{2}$.