Continuity and Differentiability

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A function f is continuous at x = c if: 1. f(c) exists (function is defined at c) 2. lim(x→c) f(x) exists 3. lim(x→c) f(x) = f(c) In other words, the limit of the function at c equals the value of th…

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Intuitively, a function is continuous at a point if you can draw its graph around that point without lifting your pen from the paper. There should be no breaks, jumps, or holes in the graph at that po…

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No, the function is not continuous at x = 0. - Left hand limit: lim(x→0⁻) f(x) = 1 - Right hand limit: lim(x→0⁺) f(x) = 2 - f(0) = 2 Since left and right hand limits are not equal, the limit at x = 0…

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If f and g are continuous functions, then: 1. (f ± g)(x) = f(x) ± g(x) is continuous 2. (f · g)(x) = f(x) · g(x) is continuous 3. (f/g)(x) = f(x)/g(x) is continuous wherever g(x) ≠ 0 Sum, difference,…

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A function f is differentiable at x = c if the derivative f'(c) exists, i.e., f'(c) = lim(h→0) [f(c+h) - f(c)]/h exists. Alternatively, f is differentiable at c if both left and right derivatives exi…

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Every differentiable function is continuous, but the converse is not true. If f is differentiable at x = c, then f is continuous at x = c. However, a function can be continuous at a point but not dif…

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If f = v ∘ u (composite function), t = u(x), and both dt/dx and dv/dt exist, then: df/dx = (dv/dt) × (dt/dx) In other words, the derivative of a composite function is the product of the derivatives …

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d/dx(sin⁻¹x) = 1/√(1-x²) This formula is valid for -1 < x < 1, which is the domain of sin⁻¹(x).