Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain. In this chapter, we will learn everything about Continuity and Differentiability of a function.
- Continuity
- Algebra of Continuous Functions
- Algebra of Derivatives
- Derivatives of Composite Functions
- Derivatives of Functions in Parametric Forms
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Mean Value Theorem
- Second Order Derivatives
FAQS on Continuity and Differentiability
Question 1: Can we say that continuity guarantees differentiability?
Answer: No, the implication of continuity is not differentiability. For example, the function ƒ: R → R defined by ƒ(x) = |x| happens to be continuous at the point 0. However, this function is not differentiable at the point 0.
Question 2: Can we say that differentiable means continuous?
Answer: Any differentiable function shall be continuous at every point that exists its domain. Furthermore, a continuous function need not be differentiable. What this means is that differentiable functions happen to be atypical among the continuous functions.
Question 3: What is the concept of limit in continuity?
Answer: A limit refers to a number that a function approaches as the approaching of the independent variable of the function takes place to a given value. For example, given the function f (x) = 3x, one can consider that the limit of f (x) as the approaching of x to 2 is 6. One can write this symbolically as f (x) = 6.
Question 4: Explain the relationship between the concept of differentiability and the concept of continuity?
Answer: The relationship between continuity and differentiability is that all differentiable functions happen to be continuous but not all continuous functions can be said to be differentiable.