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Indique las condiciones de EXISTENCIA en un punto para una función de variable real. el signo es de diferencia

The updated image indicates that we are dealing with the conditions for the existence (not just continuity) of a function at a certain point \( x_0 \). The statement in the image says:

"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: \( \lim_{x \to x_0} f(x) = f(x_0) \)."

However, your emphasis is on the conditions of **existence** (not continuity) at a point. 

### Conditions for the Existence of a Function at a Point \( x_0 \):

For a function \( f(x) \) to exist at a point \( x_0 \), the following must hold true:

1. **The Function is Defined at \( x_0 \)**: The value \( f(x_0) \) must be defined. This simply means that the function has a specific value at \( x_0 \).

2. **The Limit Exists as \( x \) Approaches \( x_0 \)**: The limit \( \lim_{x \to x_0} f(x) \) must exist. This implies that as \( x \) gets arbitrarily close to \( x_0 \), the function \( f(x) \) approaches a specific value.

3. **Equality of the Limit from Both Sides**: The limit from the left-hand side \( \lim_{x \to x_0^-} f(x) \) must equal the limit from the right-hand side \( \lim_{x \to x_0^+} f(x) \). If \( \lim_{x \to x_0^-} f(x) \neq \lim_{x \to x_0^+} f(x) \), the limit does not exist.

### Summary:
For a function to **exist** at a point \( x_0 \), the function must be defined at \( x_0 \) and the limit as \( x \) approaches \( x_0 \) from both sides must exist and be equal.

Would you like more details or have any questions?

### Related Questions:
1. What happens if the left-hand and right-hand limits at a point are not equal?
2. How is the concept of existence at a point different from continuity?
3. Can a function have an infinite limit at a point and still be considered "existent" there?
4. How do you determine the existence of a function at a point using a piecewise function?
5. What are some common cases where a function does not exist at a point?

### Tip:
When evaluating the existence of a function at a point, always check if the function is defined at that point and whether the left-hand and right-hand limits are equal and finite.

Explore the conditions of existence for real functions at a point, focusing on when a function is defined and the existence of its limit as x approaches the point. Understand the significance of left-hand and right-hand limits in determining existence.

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