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(x) continuous at x = 2?

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

Find \(\lim_{h \to 0} \frac{\sin(a+h) - \sin a}{h}\).

If \(f(x) = x^3 - 3x + 2\), find \(f'(2)\).

For the function \(f(x) = \begin{cases} kx + 1, & x \leq 1 \\ x^2, & x > 1 \end{cases}\), find the value of k such that f(x) is continuous at x = 1.

If \(f(x) = x^2\sin(\frac{1}{x})\) for \(x \neq 0\) and \(f(0) = 0\), is f differentiable at x = 0?

If \(f(x) = |x - 3|\), find the left-hand derivative and right-hand derivative at x = 3.

Which of the following functions is differentiable everywhere on \(\mathbb{R}\)?

If \(f(x)\) is continuous on \([a,b]\) and differentiable on \((a,b)\), which theorem guarantees the existence of \(c \in (a,b)\) such that \(f'(c) = \frac{f(b)-f(a)}{b-a}\)?