Continuity And Differentiability
Continuity And Differentiability are complementary to a function. For a function y = f(x), defined over a closed interval [a, b] and differentiable across the interval (a,b), there exists a point 'c' in the interval [a, b], such that it is continuous at the point x = c, if \(Lim_{x \rightarrow c}f(x) = f(c)\), and it is differntiable at the same point x = c, if \(Lim_{x \rightarrow c}f'(c) = \frac{f(c + h) - f(c)}{h}\).
Further, the function is to be first proved for its continuity at a point, before it is differentiable at the point. Let us learn more about the formulas, theorems, examples of continuity and differentiability.
What Is Continuity And Differentiability?
The continuity of a function and the differentiability of a function are complementary to each other. The function y = f(x) needs to be first proved for its continuity at a point x = a, before it is proved for its differentiability at the point x = a. The concepts of continuity and differentiability can be proved both geometrically and algebraically.
The continuity of a function f(x) at the point x = c can be proved if the limit of the function at the point is equal to the value of the function at the same point. \(Lim_{x \rightarrow c}f(x) = f(c)\). The derivative of a function y = f(x) is defined as f'(x) or d/dx.f(x) and is represented as \(f'(x) =Lim_{x \rightarrow c} \dfrac{f(x + h) - f(x)}{h}\).
Continuity Of A Function
Continuity can be simply defined for a graph y = f(x) as continuous if we are able to draw the graph easily without lifting the pencil at a point. Let f(x) be a real-valued function on the subset of real numbers and let c be a point existing in the domain of the function f(x). Then we say that the function f(x) is continuous at the point x = c if we have \(Lim_{x \rightarrow c}f(x) = f(c)\).
The continuity of a function can be explained graphically, or algebraically. In a graph the continuity of a function y = f(x) at point, is a graph line that passes continuously through the point, without any break. The continuity of a function y = f(x) can be observed algebraically if the value of the function from the left-hand limit is equal to the value of the function from the right-hand limit. \(Lim_{x \rightarrow 1^{-1}}f(x) = Lim_{x \rightarrow 1^{+1}}f(x) \). That is the values of x = 0.99, 0.998, slightly lesser than 1, has the same f(x) function value as that for x = 1.001, 1.0001, which are slightly greater than 1.
Differentiability Of A Function
The differentiation of a function gives the change of the function value with reference to the change in the domain of the function. Differentiability of a function can be understood both graphically and algebraically. Geometrically the differentiation of function is the slope of the graph of the function y = f(x) at the point x=a, in the domain of the function. Algebraically the differentiation of the function is the change in the value of the function y = f(x) from \(f(x_1)\) to \(f(x_2)\), with reference to the change in the domain value of x from \(x_1\), to \(x_2\). This can be expressed as \(\dfrac{dy}{dx} = \dfrac{f(x_2) - f(x_2)}{x_2 - x_1} \).
For a real valued function f(x) having a point x = c in the domain of this function, the derivative of the function f(x) at the point x = c is defined as \(Lim_{x \rightarrow c}f'(c) = \frac{f(c + h) - f(c)}{h}\). Thus the derivative of a function is defined as f'(x) or d/dx.f(x) and is also represented as \(f'(x) =Lim_{x \rightarrow h} \dfrac{f(x + h) - f(x)}{h}\). This process of finding the derivative is called as differentiation. Also, the phrase differentiate f(x) with respect to x is used to mean dy/dx or f'(x).
The three important rules of the algebra of differentiation of functions are as follows.
(f + g)'(x) = f'(x) + g'(x)
(f.g)'(x) = f'(x).g(x) + g'(x).f(x0
(f/g)'(x) = (f'(x).g(x) - g'(x).f(x))/(f(x))2
The following are some of the important differentiation of the functions f(x) based on the type of functions.
- Derivative of Composite Function
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Exponential Functions
- Derivatives of Logarithmic Functions
- Derivatives of Functions in Parametric Forms
Theorems on Continuity And Differentiability
The following important theorems on continuity and differentiability, set the right background for the deeper understanding of the concepts of continuity and differentiability.
Theorem 1: If two functions f(x) and g(x) are continuous at a real valued function and continuous at a point x = c, then we have:
f(x) + g(x) is continuous at the point c = c
f(x) - g(x) is continuous at a point x = c
g(x).g(x) is continuous at point x = c
f(x)/g(x) is continuous at a point x = c, provided g(c) ≠ 0
Theorem 2: For two real values functions f(x) and g(x) such that the composite function fog(x) is defined at x = c. If g(x) is continuous at x = c and the function f(x) is continuous at g(c), then fog(x) is continuous at x = c.
Theorem 3: If a given function f(x) is differentiable at a point x = c, then it is continuous at that point. This can be summarized as every differentiable function is continuous.
Theorem 4: Chain Rule: For a real valued function f(x), which is a composite of two functions u and v ie, f = vou. Also let us suppose t = u(x) and if both dt/dx and dv/dt exists, then we have df/dx = dv/dt.dt.dx.
Theorem 5: The derivative of ex with respect to x is ex. d/dx.ex = 1. And the derivative of logx with respect to x is 1/d. d/dx. logx = 1/x.
Theorem 6: (Rolle's Theorem). If a function f(x) is continuous across the interval [a, b] and differentiable across the interval (a, b), such that f(a) = f(b), and a, b are some real numbers. Then there exists a point c in the interval [a, b] such that f'(c) = 0.
Theorem 7: (Mean Value Theorem). If a function f(x) is continuous across the interval [a, b] and differentiable across the interval (a, b), then there exists a point c in the interval [a, b] such that \(f'(c) = \dfrac{f(b) - f(a)}{b - a}\).
Related Topics
The following topics help in a better understanding of Continuity And Differentiability.
FAQs on Continuity And Differentiability
What Is Continuity And Differentiability?
The continuity of a function says if the graph of the function can be drawn continuously without lifting the pencil. The differentiability is the slope of the graph of a function at any point in the domain of the function. Both continuity and differentiability, are complementary functions to each other. A function y = f(x) needs to be first continuous at a point x = a in the domain of the function before it can be proved for its differentiability.
What Is the Formula For Continuity And Differentiability?
The formulae for continuity and differentiability of a function y = f(x) at a point x = c in the domain of the function, is slightly similar. The limit of the function at x = x should be equal to the value of the function f(c), \(Lim_{x \rightarrow c}f(x) = f(c)\). Thus the derivative of a function is defined as f'(x) or d/dx.f(x) and is also represented as \(f'(x) =Lim_{x \rightarrow h} \dfrac{f(x + h) - f(x)}{h}\).
How Are Continuity And Differentiability Related?
The concepts of continuity and differentiability are complementary to each other. The function is to be first checked for continuity, for it to be differentiable at that point. The function y = f(x) cannot be differentiated at a point x = a if it is not continuous at that point.
How Do We Know If It Is Continuity Or Differentiability?
The concepts of continuity and differentiability are not different concepts but are complementary to each other. The concept of differentiability exists, only if the function is continuous at a point.
Does Differentiability Mean Continuity?
Yes. The differentiability of a function means continuity. The differentiability of a function y = f(x) at a point x = a is possible, only if it is continuous at a point x = a.