Update app.py

app.py

@@ -586,6 +586,9 @@ STRICT REQUIREMENTS:
 {SYMPY_GUIDELINES}
 9. For problems where the subject is Real Analysis, observe the following guidelines:
 
+### Real Analysis Proof Guidelines  
+For Real Analysis proofs, follow these principles to ensure clarity, rigor, and logical completeness:  
+
 a. **Justify Every Step**  
    - Provide detailed reasoning for each step and explicitly justify every bounding argument, inequality, or limit claim.  
    - If concluding that terms vanish in a limit, clearly explain why.  
@@ -597,27 +600,36 @@ b. **Handling Limits and Differentiability**
    - If verifying differentiability, explicitly state why continuity at that point is necessary and confirm that continuity has been established before proceeding.  
    - If proving continuity or differentiability, check symmetry in approach from both sides (left-hand and right-hand limits or derivatives).  
 
-c. **Function Properties and Integrability**  
+c. **Function Definitions and Explicit Statements**  
+   - When proving continuity, explicitly confirm that \( f(x) \) is **defined** at the point of interest and state what its value is.  
+   - If a function is given piecewise, clearly state the function values at transition points before evaluating limits.  
+
+d. **Limit Justifications and Transitions**  
+   - When using standard limits such as \( \lim_{t \to 0} \frac{\sin(t)}{t} = 1 \), briefly justify why it applies (e.g., from Taylor series, L'Hôpital’s Rule, or first principles).  
+   - If a limit is computed informally before a formal \(\epsilon\)-\(\delta\) proof, explicitly state that the formal proof serves to confirm the computed limit rigorously.  
+   - Ensure smooth logical transitions between different parts of the proof by briefly explaining why one step leads naturally to the next.  
+
+e. **Function Properties and Integrability**  
    - If stating that a function is Riemann integrable, compact, or uniformly continuous, explain why.  
-   - If claiming a function is continuous for all x ≠ 0, explicitly justify why using function composition, bounded functions, or known theorems.  
+   - If claiming a function is continuous for all \( x \neq 0 \), explicitly justify why using function composition, bounded functions, or known theorems.  
    - When assuming an integral is finite, provide justification based on function class properties (e.g., Riemann integrability implies boundedness).  
 
-d. **Inequalities and Asymptotics**  
+f. **Inequalities and Asymptotics**  
    - When using inequalities (e.g., Hölder’s, Jensen’s), explain why they apply and what function properties make them relevant.  
    - If using factorial ratios or infinite series sums, explicitly state their rate of convergence and reference known bounds (e.g., Stirling’s approximation).  
 
-e. **Uniform Convergence and Sequence Behavior**  
-   - When proving uniform convergence, ensure that the bound obtained is independent of x to establish uniform control.  
+g. **Uniform Convergence and Sequence Behavior**  
+   - When proving uniform convergence, ensure that the bound obtained is independent of \( x \) to establish uniform control.  
    - If using asymptotic behavior (e.g., factorial ratios tending to zero), provide explicit justification rather than just stating the result.  
 
-f. **Clarify the Use of Key Theorems (e.g., Squeeze Theorem)**  
+h. **Clarify the Use of Key Theorems (e.g., Squeeze Theorem)**  
    - When using the squeeze theorem, clearly state why both bounding functions tend to the same limit and explicitly apply the theorem in the conclusion.  
 
-g. **Logical Flow and Transitions**  
+i. **Logical Flow and Transitions**  
    - After major steps (e.g., computing a limit, verifying continuity), summarize why the step was necessary and how it connects to the next part of the proof.  
-   - Ensure smooth logical transitions between different parts of the proof by briefly explaining why one step leads naturally to the next.  
+   - If transitioning from an informal calculation to a formal proof, explicitly state the purpose of the formal proof in confirming the earlier result.  
 
-h. **Concluding and Intuitive Explanations**  
+j. **Concluding and Intuitive Explanations**  
    - Conclude with an intuitive explanation of why the result makes sense, possibly connecting it to known theorems or simple examples.  
    - In notes after the proof, highlight potential sources of confusion for students and clarify tricky aspects of the problem.  
 """