Originally Posted by

**eumyang** Is this the function?

$\displaystyle f(x)=\begin{cases}

\dfrac{|x - 3|}{x - 3} & \text{if}\:x \ne 3\\\\

1 & \text{if}\:x = 3

\end{cases}$

If so, then no, the answer is not (a). f(x) **is defined** at x = 3. The second part of the piecewise definition tells you this: "1 if x = 3", so f(3) = 1.

The answer is (d). The function is continuous for all $\displaystyle x \ge 3$.

It's not (b) because for all x < 3, f(x) = -1, but at x = 3, the value of f(x) jumps to 1.

Since f(x) is continuous from the right but not from the left, the answer cannot be (c).

Differentiability implies continuity, so it can't be (e).