Question

Let \( f(x) \) be defined as follows:

\[ f(x) = \begin{cases} 3x, & \text{if } x < 0 \\ \min(1 + x + \lfloor x \rfloor, 2 + x \lfloor x \rfloor), & \text{if } 0 \leq x \leq 2 \\ 5, & \text{if } x > 2 \end{cases} \]

where \( \lfloor . \rfloor \) denotes the greatest integer function. If \( \alpha \) and \( \beta \) are the number of points where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals:

Hint:

When evaluating discontinuity and differentiability for piecewise functions, always check transitions between piecewise segments and integer boundaries within the domain.

Answer 1

Step 1: Identify discontinuities.

The function changes definition at \(x = 0\) and \(x = 2\). Evaluate limits from left and right at these points:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} 3x = 0 \]

\[ \lim_{x \to 0^+} f(x) = \min(1 + 0 + 0, 2 + 0 \times 0) = 1 \]

\[ \lim_{x \to 2^-} f(x) = \min(1 + 2 + 1, 2 + 2 \times 1) = 4 \]

\[ \lim_{x \to 2^+} f(x) = 5 \]

Discontinuity at \(x = 0\) and \(x = 2\).

Step 2: Identify points of non-differentiability.

Check for differentiability at integer points within \([0, 2]\) and at \(x = 2\), as \(f(x)\) involves the floor function, which is non-differentiable at integers:

\[ f'(x) \text{ is not defined at } x = 1, 2 \]

Step 3: Count \(\alpha\) and \(\beta\).

\[ \alpha = 2 \quad (\text{discontinuity at 0 and 2}) \]

\[ \beta = 3 \quad (\text{non-differentiability at 0, 1, and 2}) \]

\[ \alpha + \beta = 5 \]