1. **State the problem:** We have the function $$f(x) = \frac{\lfloor x \rfloor + x}{2}$$ where $$\lfloor x \rfloor$$ is the greatest integer less than or equal to $$x$$. We want to analyze the truth of several statements about $$f$$, its limits, continuity, domain, and range. 2. **Recall the floor function properties:** - For any real number $$x$$, $$\lfloor x \rfloor$$ is the greatest integer $$\leq x$$. - If $$x$$ is an integer, $$\lfloor x \rfloor = x$$. - For non-integers, $$\lfloor x \rfloor < x < \lfloor x \rfloor + 1$$. 3. **Evaluate $$f(x)$$ for integer and non-integer values:** - If $$x = k$$ is an integer, then $$f(k) = \frac{k + k}{2} = k$$. - If $$x$$ is not an integer, write $$x = k + \delta$$ where $$k = \lfloor x \rfloor$$ and $$0 < \delta < 1$$. Then $$f(x) = \frac{k + (k + \delta)}{2} = \frac{2k + \delta}{2} = k + \frac{\delta}{2}$$. 4. **Limits approaching an integer $$k$$:** - From the left: $$x \to k^-$$ means $$x = k - \epsilon$$ with $$\epsilon \to 0^+$$. Then $$\lfloor x \rfloor = k - 1$$ because $$x$$ is just less than $$k$$. So $$f(x) = \frac{(k-1) + (k - \epsilon)}{2} = \frac{2k -1 - \epsilon}{2} = k - \frac{1}{2} - \frac{\epsilon}{2}$$. Taking limit $$\epsilon \to 0^+$$ gives $$\lim_{x \to k^-} f(x) = k - \frac{1}{2}$$. - From the right: $$x \to k^+$$ means $$x = k + \epsilon$$ with $$\epsilon \to 0^+$$. Then $$\lfloor x \rfloor = k$$. So $$f(x) = \frac{k + (k + \epsilon)}{2} = \frac{2k + \epsilon}{2} = k + \frac{\epsilon}{2}$$. Taking limit $$\epsilon \to 0^+$$ gives $$\lim_{x \to k^+} f(x) = k$$. 5. **Continuity at integers:** Since $$\lim_{x \to k^-} f(x) = k - \frac{1}{2} \neq f(k) = k$$, the function is not continuous at integers. 6. **Continuity at non-integers:** For $$x$$ not integer, $$f$$ is continuous because $$f(x) = k + \frac{\delta}{2}$$ is linear in $$\delta$$. 7. **Domain and range:** - Domain is all real numbers $$(-\infty, \infty)$$. - Range is all numbers of the form $$k + \frac{\delta}{2}$$ where $$k$$ is integer and $$0 \leq \delta < 1$$. This covers all real numbers but with jumps at integers. 8. **Check statements:** - "When $$x$$ is a positive integer, $$\lfloor x \rfloor = x$$. So $$f(x) = x$$ for all $$x$$ (not just integers) and domain and range is $$(0, \infty)$$." False: $$f(x) = x$$ only at integers, not all $$x$$; domain is all real numbers. - "The limit of $$f$$ as $$x$$ approaches integer $$k$$ from left is $$k - \frac{1}{2}$$, and $$f(k) = k$$." True. - "There are more than two correct answers, not counting this answer." We will check after. - "The limit of $$f$$ as $$x$$ approaches integer $$k$$ from right is $$k$$, and $$f(k) = k$$." True. - "When $$x$$ is a negative integer, $$\lfloor x \rfloor = -x$$. So $$f(x) = -x$$ for all $$x$$ (not just integers) and domain and range is $$(-\infty, 0)$$." False: $$\lfloor x \rfloor$$ is not $$-x$$; for negative integers $$x$$, $$\lfloor x \rfloor = x$$. - "The limit of $$f$$ as $$x$$ approaches any number other than an integer is the same as $$f(x)$$ and this implies $$f$$ is continuous for all $$x$$ including integers." False: $$f$$ is continuous at non-integers but not at integers. - "If the limit of some function $$g$$ as $$x$$ approaches integer $$k$$ is the same as $$g(k)$$ then $$g$$ is continuous at all integers." True by definition of continuity. - "There are exactly two correct answers, not counting this answer." True: the two correct statements are about the left and right limits at integers. - "There is NO correct answer, not counting this answer." False. **Final conclusion:** The two true statements are about the left and right limits at integers and the function value at those integers. **Answer:** $$\boxed{\text{The true statements are:} \\ \text{(2) The left limit at } k \text{ is } k - \frac{1}{2} \text{ and } f(k) = k, \\ \text{(4) The right limit at } k \text{ is } k \text{ and } f(k) = k, \\ \text{and (8) There are exactly two correct answers.}}$$