Concept:A function f(x) is continuous at x = a if,Left limit = Right limit = Function value = Real and finiteA function is said to be differentiable at x =a if,Left derivative = Right derivative = Well definedCalculation:Given:\(f(x) =\left | \frac{\sin2\pi x}{L}\right |\)\(\left | \frac{\sin2\pi x}{L}\right |=\frac{\sin2\pi x}{L}\) for x ? 0\(\left | \frac{\sin2\pi x}{L}\right |=-\frac{\sin2\pi x}{L}\) for x < 0.At x = 0Left limit = 0, Right limit = 0, F(0) = 0As,Left limit = Right limit = Function value = 0? \(\left | \frac{\sin2\pi x}{L}\right |\) is continuous at x = 0.Now,Left derivative (at x = 0) = -1Right derivative (at x = 0) = 1Left derivative ? Right derivative? \(\left | \frac{sin2\pi x}{L}\right |\) is not differentiable at x = 0So, option (4) is the correct answer.