**Answer: The three conditions of continuity are as follows:**

- The function is expressed at x = a.
- The limit of the function as the approaching of x takes place, a exists.
- The limit of the function as the approaching of x takes place, a is equal to the function value f(a).

The difference between the continuous and differentiable function is that the continuous function is a function, in which the curve obtained is a single unbroken curve. It means that the curve is not discontinuous. Whereas, **the function is said to be differentiable if the function has a derivative**.

**Differential Equations is an extremely important topic for the upcoming JEE Mains exam**. Important questions related to the same are made available for all the users on Vedantu's website as well as mobile applications for free. One need not pay any charges to download them.

**In calculus, a function is continuous at x = a if - and only if - all three of the following conditions are met:**

- The function is defined at x = a; that is, f(a) equals a real number.
- The limit of the function as x approaches a exists.
- The limit of the function as x approaches a is equal to the function value at x = a.

continuity, in mathematics, **rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps**. A function is a relationship in which every value of an independent variable—say x—is associated with a value of a dependent variable—say y.

A function is non-differentiable **when there is a cusp or a corner point in its graph**. For example consider the function f(x)=|x| , it has a cusp at x=0 hence it is not differentiable at x=0 .

Continuity is **the presence of a complete path for current flow**. A closed switch that is operational, for example, has continuity. A continuity test is a quick check to see if a circuit is open or closed. Only a closed, complete circuit (one that is switched ON) has continuity.

limit, mathematical concept based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.

**Answer: The three conditions of continuity are as follows:**

- The function is expressed at x = a.
- The limit of the function as the approaching of x takes place, a exists.
- The limit of the function as the approaching of x takes place, a is equal to the function value f(a).

The sum rule for derivatives states that **the derivative of a sum is equal to the sum of the derivatives**. In symbols, this means that for. f(x)=g(x)+h(x)

If a function is differentiable then it's also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it's also continuous.

For a function to be continuous at a point, **it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point**. Discontinuities may be classified as removable, jump, or infinite.

**Let's look at some:**

- Just Put The Value In. The first thing to try is just putting the value of the limit in, and see if it works (in other words substitution).
- Factors. We can try factoring.
- Conjugate.
- Infinite Limits and Rational Functions.
- L'Hôpital's Rule.
- Formal Method.

The reason is that, on one hand, **continuity is a pillar of calculus** - another being the idea of a limit - which is essential for the study of engineering and the sciences, while on the other, it has far-reaching consequences in a variety of areas seemingly unconnected with mathematics.

Real life Limits

For example, **when designing the engine of a new car, an engineer may model the gasoline through the car's engine with small intervals called a mesh**, since the geometry of the engine is too complicated to get exactly with simply functions such as polynomials. These approximations always use limits.

In Mathematics, a limit is defined as **a value that a function approaches the output for the given input values**. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity.

dx = . y2 , if y = f (x). Let f : → R be continuous on and differentiable on (a, b), such that f (a) = f (b), where a and b are some real numbers. Then there exists at least one point c in (a, b) such that f ′ (c) = 0.

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as. **f′(a−)=h→0+limh**f(a)−f(a−h)=h→0−limhf(a)−f(a−h)=x→a+lima−xf(a)−f(x) respectively.

**The limit of f at x = 3 x=3 x=3 is the value f approaches as we get closer and closer to x = 3 x=3 x=3** .

In limits, we want to get infinitely close.

x | g ( x ) g(x) g(x) |
---|---|

− 7.001 -7.001 −7.001 | 6.03 6.03 6.03 |

− 6.999 -6.999 −6.999 | 6.03 6.03 6.03 |

In calculus, a function is continuous at x = a if - and only if - all three of the following conditions are met: **The function is defined at x = a; that is, f(a) equals a real number**. The limit of the function as x approaches a exists. The limit of the function as x approaches a is equal to the function value at x = a.

The left-hand limit of a function is **the value of the function that approaches when the variable approaches its limit from the left**. This can be written as. lim x → a. f(x) = A^{–} Was this answer helpful?

Dated : 16-Jun-2022

Category : Education