Continuity, Discontinuity, and Differentiability of a Function
The conditions for continuity, discontinuity, and differentiability of a function at a point are tabulated below:
Continuity | Discontinuity | Differentiability |
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[Tex]\lim_{x \to a^{-}}f(x)=\lim_{x \to a^{+}}f(x)=f(a)[/Tex] |
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Fundamental Theorems of Continuity
- If f and g are continuous functions then f±g, fg, f/g (where, g≠0), and cf(x) all are continuous.
- If g is continuous at ‘a’ and f is continuous at g(a) then fog is also continuous at ‘a’.
- If f is continuous in its domain then |f| is also continuous in its domain
- If f is continuous in the domain D then 1/f is also continuous in D-{x:f(x) = 0}.
Fundamental Theorems of Differentiability
- If f and g are two differential functions then their sum, difference, product, and quotients are also differentiable.
- If f and g are two differential functions then fog is also differentiable.
- If f and g are not differential functions then their sum and product functions can be differential functions.
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Differential Calculus
Differential Calculus is a branch of Calculus in mathematics that is used to find rate of change of a quantity with respect to other. It involves calculating derivatives and using them to solve problems involving non constant rates of change. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications.
In this article, we have tried to provide a brief overview of the branch of Differential Calculus including topics such as limits, derivatives, various formulas for derivatives as well as application of derivatives.
Table of Content
- What is Differential Calculus?
- What is Limit?
- Limit Formulas
- Continuity, Discontinuity, and Differentiability of a Function
- Derivatives
- Differentiation Formulas
- Implicit Differentiation
- Higher Order Derivatives
- Error
- Approximation
- Inflection Point
- Tangent and Normal
- Increasing and Decreasing Function
- Maxima and Minima
- Extreme Value Theorem
- First Derivative Test
- Second Derivative Test
- Differential Equation