What is Continuity in Functions?

A function f will be continuous at a point c within its domain if it follows three conditions as mentioned below:

1. Function is defined at point c, i.e. f(c) is defined
2. Limit exists at c, i.e. $\underset{x\to c}{lim} f\left(x\right) \text{exists.}$
3. Limit is equal to value of the function $\underset{x\to c}{lim} f\left(x\right) = f\left(c\right)$

CBSE board may ask theorem-based direct questions to assess the knowledge base of students. Continuity has several theorems associated with it, including the following:

1. Intermediate Value Theorem (IVT)

Suppose a function f is continuous on closed interval [a,b] and N is any number between f(a) and f(b). In this case, at least one number c in [a,b] will exist in a way that f(c) = N.

The theorem guarantees that a continuous function on the closed interval takes on each intermediate value between function values at the endpoints. 

2. Extreme Value Theorem (EVT)

Suppose there is a function f which is continuous on closed interval [a,b]. In this case, f will attain both maximum and minimum value on that interval. EVT ensures that a continuous function on closed interval has both global maximum and global minimum. 

3. Bolzano's Theorem

Say there is a continuous function f on closed interval [a,b] and f(a) and f(b) both have opposite signs (f(a)⋅f(b)<0). In this case, there will be at least one number c in (a,b) such that f(c) = 0. This theorem is a special case of the Intermediate Value Theorem. It is used for proving the existence of roots of continuous functions.

4. Uniform Continuity Theorem

If function f is continuous on closed interval [a,b], then f is uniformly continuous on [a,b]. Uniform function is stronger condition than continuity. It is important in advanced analysis and study of function spaces.

5. Continuity of Composite Function

Say there is a continuous function f at point c and g is continuous at f(c); in this case, the composite function g∘f is continuous at c. This theorem confirms that the composition of continuous functions will also be continuous. This is useful for analysing complex functions.

6. Continuity of Inverse Functions

Suppose a function f is both continuous and bijective on an interval. In this case, the inverse of this function $f^{ -1 }$  is also continuous. This theorem confirms that the inverse of a continuous and bijective function will also be continuous. 

The NEET exam or JEE Main exam will ask problem-based questions on the continuity on interval. A function is continuous on an interval if it remains continuous at every point within that interval. Some of the examples of these functions include the following:

1. Polynomial functions: These are continuous everywhere. For example, $f \left( x \right) = x^2$ is continuous for every real number x.
2. Rational Functions: These are continuous everywhere except for the points where denominator is zero. For instance, $f \left( x \right) = \frac{1}{x}$  is continuous for all x ≠0.
3. Trigonometric Functions: Sin(x) and Cos(x) are continuous everywhere.

 

Logarithmic differentiation is a technique which is used for differentiating functions that are complex products, quotients or powers of other functions. This method takes the natural logarithm of function before differentiating, which makes it a very simple differentiation process. Let us take a look at the steps for logarithmic differentiation since questions based on this will be asked in IIT JAM exam and IISER entrance exam.

1. Start with function y = f(x) that you need to differentiate
2. Take natural logarithm of both sides ln(y) = ln(f(x))
3. Differentiate both the sides of equation with respect to x. Do remember that y is a function of x. Due to this, you will need to use the chain rule on the left side.
4. Derivative of ln(y) with respect to x is $\frac{1}{y} \cdot \frac{dy}{dx}$
5. Derivative of ln (f(x)) with respect to x is $\frac{ f^{\prime } \left(x\right) }{ f \left(x\right) }$
6. Now, once we have differentiated, let us solve for $\frac{dy}{dx}$ by multiplying y on both the sides.

 

Let us understand the importance of continuity:

• A continuous function behaves in a predictable manner without any jumps, breaks or any other erratic behaviour in the graph of the function. Due to this predictability, accurate modelling of real-world phenomena is possible.
• Many theorems like Intermediate Value Theorem and the Extreme Value Theorem depend on concept of continuity. 
• Intermediate Value Theorem depends on continuity for determining roots of equations and proving existence of solutions.
• In engineering, continuous functions are used for modelling motion, growth, decay as well as other dynamic processes. For instance, the position of moving objects is described using a continuous function of time.
• Continuity is essential for optimisation problems in the fields of economics and operations. Continuous functions help in finding maxima and minima. 
• Many numerical methods depend on the continuity of functions to approximate solutions to equations and integrals. Continuous functions can be easily approximated and analyzed using computational techniques.

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