The Continuity calculator is a helpful tool that allows for quick and efficient evaluation of the continuity of a function. This tool provides a convenient way to check whether a function is continuous at a specific point or within a given interval.

It's easy-to-understand interface and advanced features like a built-in keyboard and step-by-step process save time and effort in manual calculations.

Read and follow the given steps to use the continuity calculator.

- Enter the function you want to evaluate for continuity.
- Select the w.r.t variable.
- Type the limit of the function.
- Click
**Calculate**.

Continuity refers to the uninterrupted and smooth nature of a function. A function is considered continuous if it does not have any abrupt changes or breaks as its input values change.

In simpler terms, a continuous function can be graphed without lifting the pen from the paper, resulting in a connected curve without any holes or jumps.

To determine if a function is continuous at a specific point, we need to check three conditions:

**The function must be defined at that point:**

This means that the function has a well-defined value at the given point. If there is no value assigned to that point, continuity is not applicable.

**The limit of the function exists:**

The limit of the function as the input approaches the given point from both the left and right sides should exist and be equal. In other words, the function approaches the same value as we approach the point from either direction.

**The value of the function coincides with the limit:**

The actual value of the function at the given point should match the limit of the function as the input approaches that point. This ensures that there are no abrupt changes or jumps in the function's behavior.

If all three conditions are satisfied, the function is said to be continuous at that point. If the function is continuous at every point within its domain, it is considered a continuous function.

The concept of continuity is simple but to find it you have to test the function against three conditions that were mentioned previously. Let’s see how to do that.

- Check if the function is defined at the given point c. If f(c) is undefined or doesn't have a value, continuity is not applicable.
- Calculate the left-hand limit and the right-hand limit of the function as x approaches c.

- Evaluate lim (x → c-) f(x) by substituting x-values that approach c from the left side of c into the function.
- Evaluate lim (x → c+) f(x) by substituting x-values that approach c from the right side of c into the function.

- Compare the left-hand limit and the right-hand limit.

If lim (x → c-) f(x) = lim (x → c+) f(x), proceed to the next step. Otherwise, the function is discontinuous at point c.

- Check if the value of the function at c matches the limit. If f(c) = lim (x → c) f(x), the function is continuous at point c. If not, the function is discontinuous at point c.

**Solved Example:**

Let's consider the function f(x) = 2x + 3. We will determine if the function is continuous at the point x = 2.

**Step 1: **The function f(x) = 2x + 3 is defined for all real values of x, so it is defined at x = 2.

**Step 2: **Calculate the left-hand limit and the right-hand limit:

- lim (x → 2-) f(x) = lim (x → 2-) (2x + 3) = 2(2) + 3 = 7
- lim (x → 2+) f(x) = lim (x → 2+) (2x + 3) = 2(2) + 3 = 7

**Step 3: **The left-hand limit and the right-hand limit are both 7.

**Step 4: **Evaluate the value of the function at x = 2:

f(2) = 2(2) + 3 = 4 + 3 = 7

Since f(2) = 7 matches the limit, the function f(x) = 2x + 3 is continuous at x = 2.

In this example, the function is continuous at x = 2 because the three conditions for continuity are satisfied: the function is defined, the left-hand limit and the right-hand limit are equal, and the value of the function matches the limit.

Continuity has numerous applications in mathematics and beyond. It serves as the foundation for calculus, where concepts such as differentiation and integration heavily rely on the smoothness and connectedness of functions.

Continuity also finds applications in real analysis, differential equations, and topology, providing a framework for studying and analyzing mathematical structures.

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