Let $a, b, c$ be three real numbers. If the function $f(x) = \begin{cases} \cos(2x + \pi) & \text{if } x \leq 0 \\ ax^2 + b & \text{if } 0 < x < 1 \\ cx + 4 & \text{if } 1 \leq x \leq 2 \\ 3a + 1 & \text{if } x \geq 2 \end{cases}$ is continuous everywhere, then $b^2 - bc + c^2 =$

Let $f: R \rightarrow R$ be defined as:
$f(x) = \begin{cases} \frac{\lambda|x^{2}-5x+6|}{\mu(5x-x^{2}-6)}, & x < 2 \\ \mu, & x = 2 \\ e^{\frac{\tan(x-2)}{x-[x]}}, & x > 2 \end{cases}$
Where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x = 2$,then $\lambda + \mu$ is equal to:

Discuss the continuity of the function $f$ given by $f(x) = x^{3} + x^{2} - 1$.

Examine the continuity of the function $f(x) = 2x^{2} - 1$ at $x = 3$.

Which of the following statements is true for the graph of the function $f(x) = \log x$?

If a function $f$ defined by $f(x) = \begin{cases} \frac{1-\cos 4x}{x^2}, & x < 0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0 \end{cases}$ is continuous at $x=0$,then $a=$