Continuity and Differentiability

Introduction:
Students in Class 12 Maths are learning about Continuity and Differentiability for the first time. The complexity of this Chapter varies from moderate to challenging since difficulties may be readily answered once students have completed the formulas. However, the hardest part comes in their tests, because this Chapter has a large number of questions, the bulk of which are in the form of application kinds.If the function is defined at x = c .Because the function's value at x = c equals the function's limit at x = c, the function is continuous at x = c. If function f is not continuous at c, then c has termed a point of discontinuity of function f. The feature of continuity may be seen in many aspects of nature. There is a continual flow of water in the rivers. Because human existence is a continuous passage of time, you will constantly get older. The list goes on. Similarly, we have the idea of function continuity in mathematics. Simply said, a function's curve is deemed continuous if you can write it on a graph without raising your pen even once (provided that you can draw well). Actually, it's a really straightforward and true definition. However, for the sake of advanced mathematics, we must define it more accurately.

List of topics according to NCERT and JEE Main/NEET syllabus:

• Algebra of a continuous function
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second-Order Derivative
• Mean Value Theorem
  
  

Important concepts and Laws:

The definition of the term "continuity" is the same as what we use in our everyday lives. The water flow in rivers, for example, is constant. The passage of time in human existence is constant, i.e. we continue to age and so on. Similarly, in mathematics, we have the concept of function continuity.

When we claim that a function f(x) is continuous at x = a, we imply that the graph of the function has no holes or gaps at the point (a, f(a)). In basic terms, a function is considered to be continuous if its curve can be sketched on a graph without ever raising your pen.

Let c be a point in the domain of f and f be a real function on a subset of the real numbers.

Lim x → c f(x) = f(c)

Let f and g be two continuous real functions at a real integer c.

• At x = c, f + g is continuous.
• At x = c, f - g is continuous.
• At x = c, f * g is continuous.
• At x = c, where g(c) 0, f/g is continuous.
  
  

A one-real-variable differentiable function is one whose derivative exists at each point in its domain. As a result, each point in the domain of a differentiable function's graph must have a tangent line. There should be no fractures, bends, or cusps, and it should be rather smooth.

The theorem of Mean Value

In Calculus, there are two fundamental results. Both will be discussed, as well as the geometric interpretations of these theorems.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology.

Theorem of Rolle:

Let f be a real-valued function that is defined on the closed interval [a, b] and has the property that

On the closed interval [a, b], it is continuous.

On the open interval, it is differentiable (a, b)

f(a) = f(b) = f(c) = f(d) = f(e) = f( (b)

Then a real integer c (a, b) exists such that f′(c) = 0.

Inverse Trigonometric Functions: The class of inverse functions is quite broad, and it is responsible for accomplishing the inverse of what a function performs, as the name implies. The multiplication function, for example, is the inverse of the division function. Because of their broad relevance, it's critical to comprehend their continuous and differentiable character across domains.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.
  
  

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 12 Physics.
• NCERT exemplar solutions for class 12 Chemistry.
• NCERT exemplar solutions for class 12 Mathematics.
• NCERT exemplar solutions for class 12 Biology.