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To determine the value of \( c \) that makes the function \( f(x) \) continuous at \( x = 0 \), we need to ensure that the limit of \( f(x) \) from both sides of \( x = 0 \) is equal and that \( f(0) \) is well-defined.

The function is defined as:

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

### Conditions for Continuity:
For \( f(x) \) to be continuous at \( x = 0 \), the following must hold:
1. \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \)
2. \( f(0) \) must equal this common limit.

### Step 1: Evaluate \( \lim_{x \to 0^-} f(x) \)
For \( x < 0 \), we have:
\[
f(x) = e^x \sin(2\pi x)
\]
As \( x \to 0^- \), \( e^x \to 1 \) and \( \sin(2\pi x) \to 0 \). Therefore:
\[
\lim_{x \to 0^-} f(x) = 1 \times 0 = 0
\]

### Step 2: Evaluate \( \lim_{x \to 0^+} f(x) \)
For \( x \geq 0 \), we have:
\[
f(x) = x^2 - cx
\]
At \( x = 0 \), the function becomes:
\[
f(0) = 0^2 - c \times 0 = 0
\]
Thus, \( f(0) = 0 \). Now, we calculate the limit as \( x \to 0^+ \):
\[
\lim_{x \to 0^+} (x^2 - cx) = 0^2 - c \times 0 = 0
\]

### Step 3: Ensure Limits Match
Since:
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 0
\]
The function is already continuous for any value of \( c \).

### Conclusion:
The function \( f(x) \) is continuous at \( x = 0 \) for **any value of \( c \)**. Thus, the correct option is:

\[
\boxed{\text{any value of } c}
\]

---

Would you like further clarification or details? Here are some related questions for deeper understanding:

1. How do you verify continuity for piecewise functions?
2. What happens if the left-hand and right-hand limits do not match?
3. How would the continuity conditions change if the function had a jump discontinuity?
4. Can you give an example where a function is continuous but not differentiable at a point?
5. How do different trigonometric components affect limits near zero?

**Tip:** Always check both left-hand and right-hand limits to ensure continuity at a point, especially for piecewise functions.

Consider the function f(x) defined as: f(x) = {e^x * sin(2πx), x < 0; x^2 - cx, x >= 0}. Choose the option that describes all values of c such that f(x) is continuous at x = 0.

Solve the continuity problem for a piecewise function f(x) with exponential and quadratic terms. Learn how to find values of c such that the function is continuous at x = 0. This problem covers limits and continuity of functions.

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