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Indique las condiciones de EXISTENCIA en un punto para una función de variable real

The problem you uploaded is asking for the conditions under which a function \( f \) is continuous at a point \( x_0 \). The statement from the image reads:

"La función \( f \) es continua en el punto \( x_0 \) si, y solo si, \( f(x) \) está definida y existe \( \lim_{x \to x_0} f(x) \) tal que se verifica: \( \lim_{x \to x_0} f(x) = f(x_0) \)."

This is a formal definition of continuity at a point for a function of a real variable.

### Conditions for Continuity at a Point \( x_0 \):

1. **Existence of the Function at \( x_0 \)**: The function \( f(x) \) must be defined at \( x_0 \). That is, \( f(x_0) \) exists.
   
2. **Existence of the Limit**: The limit of the function as \( x \) approaches \( x_0 \) must exist. This means \( \lim_{x \to x_0} f(x) \) exists.
   
3. **Equality of the Limit and Function Value**: The value of the limit as \( x \) approaches \( x_0 \) must equal the function value at \( x_0 \). That is, \( \lim_{x \to x_0} f(x) = f(x_0) \).

### Summary:
For a function \( f(x) \) to be continuous at a point \( x_0 \), it must meet the following conditions:

- \( f(x_0) \) exists.
- \( \lim_{x \to x_0} f(x) \) exists.
- \( \lim_{x \to x_0} f(x) = f(x_0) \).

Would you like more details or have any questions?

### Related Questions:
1. What is the difference between continuity and differentiability at a point?
2. How can we check the continuity of a piecewise function?
3. Can a function be discontinuous at a point where the limit exists? Explain.
4. What are removable discontinuities and how are they related to continuity?
5. How does the concept of continuity apply to multivariable functions?

### Tip:
To check the continuity of a function graphically, observe whether the graph has any breaks, holes, or jumps at the point in question.

Understand the conditions of existence for a real variable function at a point with this detailed explanation. Learn about the mathematical concept of continuity, including the requirements for a function to be continuous at a specific point.

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