If $f(x) = [x]$ for $x \in (-1, 2)$,then $f$ is discontinuous at (where $[x]$ represents the floor function).

If $f(x) = \begin{cases} \frac{8^x-4^x-2^x+1}{x^2}, & \text{if } x > 0 \\ e^x \sin x + x + \lambda \log 4, & \text{if } x \leqslant 0 \end{cases}$ is continuous at $x = 0$,then the value of $1000 e^\lambda = $

If $f(x) = \begin{cases} x^2, & x \ne 1 \\ 2, & x = 1 \end{cases}$,then which of the following is true?

Let $f(x) = \frac{g(x)}{h(x)}$,where $g$ and $h$ are continuous functions on the open interval $(a, b)$. Which of the following statements is true for $a < x < b$?

If $f(x) = \begin{cases} ax^2 + bx + 1 & \text{if } |2x - 3| \geq 2 \\ 3x + 2 & \text{if } \frac{1}{2} < x < \frac{5}{2} \end{cases}$ is continuous on its domain,then $a + b$ has the value:

If the function $f(x) = \begin{cases} [\tan(\frac{\pi}{4} + x)]^{\frac{1}{x}}, & x \neq 0 \\ K, & x = 0 \end{cases}$ is continuous at $x = 0$,then $K = ?$