The composition of two continuous function is a continuous function.

Explanation

To determine whether the statement "The composition of two continuous functions is a continuous function" is true or false, we will prove that it is indeed true.

### Step-by-Step Solution:

1. **Define Continuous Functions**:
Let \( f(x) \) and \( g(x) \) be two continuous functions. By definition, a function \( f(x) \) is continuous at a point \( c \) if:
\[ \lim_{x \to c} f(x) = f(c) \]
This means that the left-hand limit (LHL), right-hand limit (RHL), and the function value at that point must all be equal.

2. **Consider the Composition**:
We want to prove that the composition \( h(x) = f(g(x)) \) is continuous.

3. **Choose a Point**:
Let \( k \) be any point in the domain of \( g \) such that \( g(k) \) is in the domain of \( f \).

4. **Evaluate Limits**:
Since \( g(x) \) is continuous at \( k \), we have:
\[ \lim_{x \to k} g(x) = g(k) \]

5. **Apply Continuity of \( f \)**:
Since \( f(x) \) is continuous at \( g(k) \), we can write:
\[ \lim_{y \to g(k)} f(y) = f(g(k)) \]
where \( y = g(x) \).

6. **Combine the Limits**:
Now, we can express the limit of the composition:
\[ \lim_{x \to k} h(x) = \lim_{x \to k} f(g(x)) = f\left(\lim_{x \to k} g(x)\right) = f(g(k)) \]

7. **Conclusion**:
Therefore, we have shown that:
\[ \lim_{x \to k} h(x) = h(k) \]
This proves that \( h(x) = f(g(x)) \) is continuous at \( k \).

8. **Generalization**:
Since \( k \) was arbitrary, we conclude that the composition of two continuous functions \( f \) and \( g \) is continuous everywhere in their domains.

### Final Statement:
Thus, the statement "The composition of two continuous functions is a continuous function" is **true**.

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