BackContinuity and Differentiability in Business Calculus
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Continuity and Differentiability
Continuity of Functions
Continuity is a fundamental concept in calculus, describing whether a function behaves smoothly without breaks or jumps at a given point or over an interval.
Definition: A function f(x) is continuous at x = a if the following three conditions are met:
f(a) is defined.
limx→a f(x) exists.
limx→a f(x) = f(a)
Types of Continuity:
Right-continuous at x = a: limx→a+ f(x) = f(a)
Left-continuous at x = a: limx→a- f(x) = f(a)
Discontinuity: Occurs when any of the above conditions fail. Common types include jump, infinite, and removable discontinuities.
Example: Consider the piecewise function:
For x < 3, f(x) = x^2
For x ≥ 3, f(x) = 2x + 1
To check continuity at x = 3:
f(3) = 2(3) + 1 = 7
limx→3- f(x) = 3^2 = 9
limx→3+ f(x) = 7
Since limx→3- f(x) ≠ limx→3+ f(x), the function is not continuous at x = 3.
Differentiability of Functions
Differentiability describes whether a function has a well-defined tangent (slope) at a point. A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability.
Definition: f(x) is differentiable at x = a if the following limit exists:
Relationship: If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
Non-differentiability: Occurs at points where the function has a sharp corner, cusp, or vertical tangent.
Example: The absolute value function f(x) = |x| is continuous everywhere but not differentiable at x = 0 due to the sharp corner.
Instantaneous Rate of Change and Tangent Lines
The derivative of a function at a point represents the instantaneous rate of change, which is also the slope of the tangent line to the function at that point.
Definition: The instantaneous rate of change of f(x) at x = a is:
Tangent Line Equation: The equation of the tangent line to y = f(x) at x = a is:
Interpretation: The derivative gives the best linear approximation to the function near x = a.
Example: For f(x) = x^2 at x = 2:
f(2) = 4
f'(x) = 2x, so f'(2) = 4
Tangent line:
Summary Table: Continuity vs. Differentiability
Property | Continuity | Differentiability |
|---|---|---|
Definition | Function has no breaks or jumps at a point | Function has a well-defined tangent (slope) at a point |
Required for | Differentiability | Applications involving rates of change |
Common Failures | Jump, removable, infinite discontinuity | Sharp corner, cusp, vertical tangent |
Example | f(x) = |x| is continuous everywhere | f(x) = |x| is not differentiable at x = 0 |
Additional info:
These concepts are foundational for understanding optimization, marginal analysis, and modeling in business calculus.
Continuity and differentiability are prerequisites for applying calculus techniques to real-world business problems, such as cost minimization and revenue maximization.
