The given statement is in Chinese and translates to:

1. Determine whether the following statement is true or false: If ( f: \mathbb{R}^n \rightarrow \mathbb{R} ) is differentiable at point ( x ), then the function is continuous at point ( x ).

Let's analyze this mathematically.翻译成中文

Statement Analysis

To determine whether the statement is true or false, let's recall the definition and relationship between differentiability and continuity:

• Differentiability: A function ( f ) is said to be differentiable at a point ( x ) if the derivative ( f'(x) ) exists at that point.
• Continuity: A function ( f ) is said to be continuous at a point ( x ) if ( \lim_{{h \to 0}} f(x+h) = f(x) ).

Relationship between Differentiability and Continuity

It is a well-known fact in calculus that if a function is differentiable at a point, then it must be continuous at that point. This means that differentiability implies continuity. However, the converse is not necessarily true; a function can be continuous at a point without being differentiable there.

Conclusion

Given that differentiability implies continuity, the statement "If ( f: \mathbb{R}^n \rightarrow \mathbb{R} ) is differentiable at point ( x ), then the function is continuous at point ( x )" is true.

Would you like more details or have any questions about this? Here are some related questions to consider:

1. What is the formal definition of differentiability in ( \mathbb{R}^n )?
2. How can we prove that differentiability implies continuity?
3. Can you provide an example of a function that is continuous but not differentiable?
4. What are the conditions for a function to be differentiable in multiple dimensions?
5. How does the concept of differentiability extend to complex-valued functions?
6. What is the difference between partial differentiability and total differentiability?
7. Are there any exceptions to the rule that differentiability implies continuity?
8. How does the differentiability of a function affect its behavior near the point of differentiability?

Tip: When analyzing mathematical statements, always start by recalling the definitions and fundamental theorems related to the concepts involved. This helps in constructing a logical and accurate argument.