NEST2026MathematicsContinuity and Differentiability
Let f : R R be a function defined by f(x) = cases x ( e^ 1/x - e^ -1/x e^ 1/x + e^ -1/x ) & if x 0 0 & if x = 0. cases Then
Options
- Athere exists a constant C such that |f(x)| C for all x R .
- Bf is monotonically increasing in the interval (-1, 1) .
- Cf is not continuous at x = 0 .
- Df is differentiable at x = 0 .
Correct answer
A. there exists a constant C such that |f(x)| C for all x R .
Step-by-step solution
The given function can be rewritten as f(x) = x ( 1 x ) for x 0 , and f(0) = 0 . Let us check the continuity of f at x = 0 : _ x 0 f(x) = _ x 0 x ( 1 x ) Since | ( 1 x ) | As x 0 , |f(x)| 0 , which means _ x 0 f(x) = 0 = f(0) . Thus, f is continuous at x = 0 . Let us check the differentiability of f at x = 0 : Right-hand derivative: f'(0^+) = _ h 0^+ h (1/h) - 0 h = _ h 0^+ ( 1 h ) = 1 Left-hand derivative: f'(0^-) = _ h 0^- h (1/h) - 0 h = _ h 0^- ( 1 h ) = -1 Since f'(0^+) f'(0^-) , f is not differentiable at x =