Ch 05 - Continuity and Differentiability12th Mathematics
Condition for Continuity at a Point
What is the strict mathematical condition for a function to be continuous at a specific point ?
What is the strict mathematical condition for a function to be continuous at a specific point ?
A function is continuous at a point in its domain if and only if:
- Left-Hand Limit (LHL) = Right-Hand Limit (RHL) = Value of the function at c
- Mathematically:
- The graph of can be drawn through the point without lifting the pen.
Boundary Conditions / Domain
- The limit must evaluate to a finite real number.
- The point must lie in the domain of .
Prism Memory Hook
Limit equals Value (L=V)
Removable vs. Jump Discontinuity
What is the mathematical difference between a removable discontinuity and a jump discontinuity?
What is the mathematical difference between a removable discontinuity and a jump discontinuity?
Removable Discontinuity
The limit exists (LHL = RHL), but it is NOT equal to the function's value . It represents a fixable hole in the graph.
Jump Discontinuity
The LHL and RHL both exist as finite values but are unequal. The graph jumps from one y-value to another.
Boundary Conditions / Domain
- Both cases require the one-sided limits to be finite. If a limit is infinite, it is an Infinite Discontinuity.
Prism Memory Hook
Removable = Hole, Jump = Gap
Standard Continuous Families
Identify the natural domains over which polynomial, exponential, modulus, and logarithmic functions are universally continuous.
Identify the natural domains over which polynomial, exponential, modulus, and logarithmic functions are universally continuous.
The following standard families of functions are continuous everywhere on their respective natural domains:
- Polynomials, Constants, Modulus , , , : Continuous on all of .
- Logarithm (): Continuous on .
Boundary Conditions / Domain
- for polynomials, exponentials, sine, cosine.
- strictly for logarithmic functions.
Prism Memory Hook
A premium mnemonic for this topic is available in the Prism App.
Also in this chapter
49 more cards to study in Prism
- 4Continuity Boundaries of Rational Trig Functions
- 5Algebra of Continuous Functions
- 6Continuity of Composite Functions
- 7Discontinuities of the Greatest Integer Function
- 8Oscillatory Discontinuity Edge Case
- 9Nullifying Discontinuity via Multiplication
- 10First Principle of Derivatives
- 11Left and Right Hand Derivatives (LHD & RHD)
- 12Relationship Between Continuity and Differentiability
- 13Non-Differentiability of the Modulus Function
- 14Constructing Multi-Point Non-Differentiable Functions
- 15Higher Order Differentiability of Modulus Powers
- 16Product Rule of Differentiation
- 17Quotient Rule of Differentiation
- 18Chain Rule for Nested Functions
- 19Derivatives of Inverse Sine and Cosine
- 20Derivatives of Inverse Tangent and Cotangent
- 21Derivatives of Inverse Secant and Cosecant
- 22Derivative of Exponential Functions
- 23Derivative of Logarithmic Functions
- 24Second-Order Derivative Notation
- 25Derivative of a Function w.r.t. Another Function
- 26Pre-Substitution for Inverse Trigonometry: Identification
- 27Pre-Substitution for Inverse Trigonometry: Execution SOP
- 28Implicit Differentiation: Identification
- 29Implicit Differentiation: Execution SOP
- 30Archetypes for Logarithmic Differentiation
- 31Logarithmic Differentiation: Log Application Phase
- 32Logarithmic Differentiation: Implicit Isolation Phase
- 33Logarithmic Product-to-Sum Transformation
- 34Change of Base Algorithm for Differentiation
- 35First Order Parametric Differentiation
- 36Second Order Parametric Derivative Trap
- 37Infinite Repeating Roots: Algebraic Setup Phase
- 38Infinite Repeating Roots: Implicit Execution Phase
- 39Differentiation of Matrix Determinants
- 40Implicit Trig Inversion Trick (HOTS): Isolation Phase
- 41Implicit Trig Inversion Trick (HOTS): Inversion Phase
- 42Proving Second Order DEs (HOTS): Quotient Rule Bypass
- 43Proving Second Order DEs (HOTS): Implicit Execution
- 44Rolle's Theorem Strict Conditions
- 45Lagrange's Mean Value Theorem (MVT)
- 46Geometric Interpretation: Rolle's vs MVT
- 47Inequality Proofs via MVT: Setup Phase
- 48Inequality Proofs via MVT: Bounding Phase
- 49Intermediate Value Theorem & Root Existence
- 50L'Hôpital's Rule for Indeterminate Limits
- 51Asymptotic Growth: Exponential vs Polynomial
- 52Radius of Curvature Invariant for a Circle
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