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To solve this problem, we need to translate the given argument into symbolic form using predicates, and then determine whether the argument is valid or invalid.

### Step 1: Define the predicates
Let’s define the following predicates:

- \( C(x) \): \( x \) is a continuous function.
- \( D(x) \): \( x \) is a differentiable function.
- Let \( f(x) = |x| \) represent the function in question.

### Step 2: Translate the premises (P1 and P2) and conclusion

1. **P1**: "All continuous functions are also differentiable functions."
   - This is a universal statement. We can translate it as:
     \[
     \forall x \, (C(x) \rightarrow D(x))
     \]
     This means: "For all \( x \), if \( x \) is continuous, then \( x \) is differentiable."

2. **P2**: "The function \( f(x) = |x| \) is not a differentiable function."
   - This is a specific statement about the function \( f(x) = |x| \). So, we can write:
     \[
     \neg D(f)
     \]
     This means: "The function \( f \) is not differentiable."

3. **Conclusion**: "The function \( f(x) = |x| \) is not a continuous function."
   - This is the conclusion we are trying to verify. In symbolic form:
     \[
     \neg C(f)
     \]
     This means: "The function \( f \) is not continuous."

### Step 3: Check the validity of the argument

- **Form of the argument**: 
  - P1 states \( \forall x (C(x) \rightarrow D(x)) \), which means "If \( f \) is continuous, then it must be differentiable."
  - P2 asserts \( \neg D(f) \), meaning \( f \) is not differentiable.
  - From these two premises, using the contrapositive of P1 (\( \neg D(f) \rightarrow \neg C(f) \)), we can logically conclude that \( f \) is not continuous.

This is a valid **contrapositive form** of reasoning. The argument is valid because if \( f \) is not differentiable (as stated in P2), then according to P1, it cannot be continuous.

### Conclusion: The argument is **valid** based on the contrapositive form of logic.

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Would you like more details on this solution? Let me know if you have any questions!

### Related Questions:
1. How do we identify the contrapositive in logical arguments?
2. Why is the function \( f(x) = |x| \) not differentiable at \( x = 0 \)?
3. Can a function be continuous but not differentiable? Provide examples.
4. What are other common logical forms (e.g., modus ponens, modus tollens) in predicate logic?
5. What is the difference between universal and existential quantifiers in logical statements?

### Tip:
In logic, the contrapositive of a statement \( P \rightarrow Q \) is \( \neg Q \rightarrow \neg P \), and it is always logically equivalent to the original statement.

Use predicates, defining all letters used, and translate the following argument into symbols. Use the valid universal forms or the invalid universal forms to determine whether the argument is valid or invalid. State which form you are using.

Explore how to use predicate logic to analyze the relationship between continuous and differentiable functions. Learn how to determine if a function is continuous based on its differentiability using contrapositive logic.

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