Continuity and Differentiability

If a function f(x) is continuous at x = a, which of the following must be true?

Consider the function \(f(x) = \begin{cases} x^2, & x \neq 2 \\ 5, & x = 2 \end{cases}\). Is f continuous at x = 2?

If \(f(x) = |x|\), which of the following is true about differentiability at x = 0?

If \(f(x) = \begin{cases} kx + 1, & x \leq 1 \\ x^2, & x > 1 \end{cases}\) is continuous at x = 1, find k.

Which of the following statements is TRUE?

Find the derivative of \(f(x) = 3x^3 - 5x + 2\) at x = 1.

For which value of a is the function \(f(x) = \begin{cases} x^2 - a, & x \leq 2 \\ 3x - 2, & x > 2 \end{cases}\) continuous at x = 2?

If \(f(x) = x^{1/3}\), then at x = 0,

Which of the following functions is NOT differentiable at x = 0?

If \(f(x) = \begin{cases} x^2 + 2x, & x \neq 1 \\ k, & x = 1 \end{cases}\) is continuous and differentiable at x = 1, find k.