Let f(x) = [x] |x^2 - a^2| be defined on the interval (-2, 2) , where a is a real number in (0, 2) and [x] denotes the greatest integer function. If f(x) is not differentiable at exactly 3 points in (-2, 2) , then the value of 10a is ________.

Step-by-step solution

The possible points of non-differentiability in (-2, 2) are the integers x = -1, 0, 1 (where [x] jumps) and the roots of the modulus x = a, -a . At x = 0 : LHL = _ x 0^- (-1)|x^2 - a^2| = -a^2 . RHL = _ x 0^+ (0)|x^2 - a^2| = 0 . Since a (0, 2) , -a^2 0 . Thus, f(x) is discontinuous and non-differentiable at x = 0 for all valid a . At x = 1 : LHL = _ x 1^- (0)|x^2 - a^2| = 0 . RHL = _ x 1^+ (1)|x^2 - a^2| = |1 - a^2| . For continuity, we need |1 - a^2| = 0 a = 1 . If a 1 , f(x) is discontinuous (and non-differentia