Continuity and Differentiability
Gujarat Board · Class 12 · Mathematics
Quick revision notes for Continuity and Differentiability — Gujarat Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Continuity of Functions
- A function f(x) is continuous at point c if lim(x→c) f(x) = f(c)
- For continuity at c: left-hand limit = right-hand limit = function value at c
- Graphically, continuous functions can be drawn without lifting the pen
Differentiability
- A function is differentiable at c if lim(h→0) [f(c+h) - f(c)]/h exists
- Every differentiable function is continuous, but not vice versa
- Differentiability implies continuity but continuity doesn't guarantee differentiability
Chain Rule
- Used to differentiate composite functions f(g(x))
- If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx)
- Essential for differentiating complex functions
Derivatives of Inverse Trigonometric Functions
- Inverse trigonometric functions have specific derivative formulas
- Domain restrictions are important for these functions
- Often combined with chain rule for composite functions
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