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We are given the piecewise function:

\[
f(x) = 
\begin{cases} 
x + e^x, & x < 0 \\
x^2, & x \geq 0
\end{cases}
\]

We need to determine if this function is continuous or discontinuous at the points \(x = 0\) and \(x = 4\).

### 1. Continuity at \(x = 0\)

A function is continuous at \(x = 0\) if:
- \(f(0)\) is defined.
- \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x)\).
- \(\lim_{x \to 0} f(x) = f(0)\).

#### Checking these conditions:
- \(f(0) = 0^2 = 0\) (using the second case of the piecewise function).
- The left-hand limit \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + e^x)\). Since \(e^x \to 1\) as \(x \to 0\), we get:
  \[
  \lim_{x \to 0^-} (x + e^x) = 0 + 1 = 1.
  \]
- The right-hand limit \(\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0^2 = 0\).

Since \(\lim_{x \to 0^-} f(x) = 1\) and \(\lim_{x \to 0^+} f(x) = 0\), the limits are not equal. Therefore, the function is **discontinuous** at \(x = 0\).

### 2. Continuity at \(x = 4\)

Here, \(x = 4\) falls within the second piece of the function, so we just check the limit at \(x = 4\). The function is defined by \(f(x) = x^2\) when \(x \geq 0\), so:
- \(f(4) = 4^2 = 16\).
- The limit \(\lim_{x \to 4} f(x) = \lim_{x \to 4} x^2 = 16\).

Since the limit exists and equals \(f(4)\), the function is **continuous** at \(x = 4\).

### Conclusion:
- The function is **discontinuous** at \(x = 0\).
- The function is **continuous** at \(x = 4\).

Would you like more details or further clarification?

Here are 5 related questions for further exploration:
1. How do we determine the differentiability of a piecewise function at a given point?
2. Can a function be continuous but not differentiable at a point? If so, what are the conditions?
3. What happens to the limits if we modify the function for \(x < 0\) to be \(x^2 + e^x\)?
4. How would the continuity of \(f(x)\) change if we defined a different value for \(f(0)\)?
5. How can we visualize the discontinuity at \(x = 0\) on a graph?

**Tip:** Always check both the left-hand and right-hand limits separately when analyzing the continuity of piecewise functions at boundary points.

Determine if the function f(x) = { x + e^x, x < 0; x^2, x >= 0 } is continuous or discontinuous at the points x = 0 and x = 4.

This page discusses how to check the continuity of a piecewise function defined as f(x) = { x + e^x, x < 0; x^2, x >= 0 } at specific points, x = 0 and x = 4. The solution includes analysis of limits and continuity conditions.

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