Let `phi:[0,1]->[0,1]` be a continuous and one-one function. Let `phi(0)=0,phi(1)=1,phi(1/2)=p and phi(1/4)=q` then

Options

A. `p gt q`
B. `p lt q`
C. p = q
D. No comparison between p and q

Answer

To solve the problem, we need to analyze the properties of the function \( \phi \) defined from the interval \([0, 1]\) to itself. Here are the steps to arrive at the conclusion regarding the relationship between \( p \) and \( q \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We have a function \( \phi: [0, 1] \to [0, 1] \) which is continuous and one-to-one. This means that for every \( x_1, x_2 \in [0, 1] \), if \( x_1 < x_2 \), then \( \phi(x_1) < \phi(x_2) \). Therefore, \( \phi \) is a strictly increasing function.

**Hint**: Recall that a one-to-one function (injective) means that different inputs yield different outputs.

2. **Given Values**:
We know:
- \( \phi(0) = 0 \)
- \( \phi(1) = 1 \)
- \( \phi\left(\frac{1}{2}\right) = p \)
- \( \phi\left(\frac{1}{4}\right) = q \)

3. **Comparing Inputs**:
Since \( \frac{1}{2} > \frac{1}{4} \), and knowing that \( \phi \) is strictly increasing, we can conclude:
\[ \phi\left(\frac{1}{2}\right) > \phi\left(\frac{1}{4}\right) \]

4. **Substituting Known Values**:
Substituting the values we have:
\[ p > q \]

5. **Conclusion**:
Therefore, the relationship between \( p \) and \( q \) is:
\[ p > q \]

### Final Answer:
The relationship between \( p \) and \( q \) is \( p > q \).