The function $f(x) = [x] + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function respectively, is discontinuous at
Step-by-Step Solution
Key Concept: At any point x = n (integer), [x] = n and {x} = 0, so f(n) = n + √0 = n. For x in (n, n+1), [x] = n and {x} = x - n, giving f(x) = n + √(x-n), which is continuous and equals n at x = n. The left and right limits both equal n, ensuring continuity everywhere.
The graph of $f(x)$ shown illustrates a function that increases monotonically from the left, passing through points at $x = 1, 2, 3$. The curve exhibits increasing slope, suggesting the derivative increases as $x$ increases, consistent with a convex function.
Correct Answer: 3