Let me explain the three conditions of continuity in a more detailed and personal way.

1. The limit must exist at that point: When we say the limit exists, we mean that as the x-values approach a certain point, the y-values also approach a specific value. In simpler terms, the function should not have any abrupt jumps or holes at that particular point. It should smoothly approach a certain value as you get closer and closer to that point.

To give you an example, let’s consider the function f(x) = x^2. If we take the limit as x approaches 1, we can see that the y-values approach 1 as well. There are no sudden jumps or gaps in the graph of this function, making it continuous at x = 1.

2. The function must be defined at that point: For a function to be continuous at a specific point, it must be defined at that point. This means that the function should have a meaningful output for the given input. In other words, there should be no undefined or missing values at that particular point.

Taking the same example of f(x) = x^2, the function is defined for all real numbers. So, it is defined at x = 1 as well. However, if we consider a function like g(x) = 1/x, we can see that it is not defined at x = 0. Therefore, g(x) is not continuous at x = 0.

3. The limit and the function must have equal values at that point: This condition ensures that the function is smooth and consistent at the specific point. It means that the limit value and the actual value of the function should be the same at that point. In mathematical terms, this can be expressed as the limit of the function as x approaches a should be equal to the value of the function at x = a.

Let’s take an example to illustrate this condition. Consider the function h(x) = 2x + 3. If we take the limit as x approaches 2, we get 7. And if we substitute x = 2 into the function, we also get 7. Hence, the limit and the function have equal values at x = 2, satisfying this condition of continuity.

To summarize, for a function to be continuous at a point, three conditions must be met: the limit must exist at that point, the function must be defined at that point, and the limit and the function must have equal values at that point. These conditions ensure that the function is smooth, without any abrupt changes or undefined points, providing a continuous and connected graph.