If f(x)= array cc x & , 0 x 1 2 x-1 & , x>1 array . then

Step-by-step solution

To check continuity at x=1 , we evaluate the left-hand limit and right-hand limit. Left-hand limit: _ x 1⁻ f(x) = _ x 1⁻ x = 1 . Right-hand limit: _ x 1⁺ f(x) = _ x 1⁺ (2x - 1) = 2(1) - 1 = 1 . Since f(1) = 1 and the limits are equal, f(x) is continuous at x=1 . To check differentiability at x=1 , we evaluate the left-hand derivative and right-hand derivative. Left-hand derivative: f'(1⁻) = d dx (x) = 1 . Right-hand derivative: f'(1⁺) = d dx (2x - 1) = 2 . Since f'(1⁻) f'(1⁺) , the function is not differentiable at