Update app.py

app.py

@@ -592,6 +592,10 @@ STRICT REQUIREMENTS:
     - When using supremum/infimum, explain why it behaves as expected under limits, differentiation, or integration. Provide explicit justification that the supremum argument does not introduce discontinuities.
     - If you conclude certain terms vanish in a limit, clearly justify why
     - If you state that a function has a certain property, such as being Riemann integrable or compact or uniformly continuous for example, clearly explain why
+    - When using limit substitutions, explicitly justify why the transformation is valid and preserves the limit structure.
+    - If verifying differentiability, explicitly state why continuity at that point is necessary and how it connects to the derivative’s existence."
+    - When proving differentiability or continuity, confirm whether the function behaves symmetrically for positive and negative values approaching the given point. If necessary, compute left-hand and right-hand derivatives."
+    - After completing a major step (e.g., computing a limit or verifying continuity), briefly explain why that step matters in the overall proof structure."
     - Conclude with a brief intuitive explanation of why the result makes sense, possibly by connecting it to known theorems or simple examples.
     - In notes after the proof, if you observe aspects of the problem that might confuse students, address them."""