If f and g are both continuous at x = a, then f-g is

Options

A. discontinuous at x = a
B. not defined
C. continuous at x = a
D. none of these

Answer

To determine whether \( f - g \) is continuous at \( x = a \) given that \( f \) and \( g \) are both continuous at \( x = a \), we can follow these steps:

### Step 1: Understand the Definition of Continuity
A function \( h(x) \) is continuous at \( x = a \) if:
\[ \lim_{x \to a} h(x) = h(a) \]
This means that the limit of the function as \( x \) approaches \( a \) must equal the value of the function at \( a \).

### Step 2: Apply the Continuity of \( f \) and \( g \)
Since \( f \) is continuous at \( x = a \), we have:
\[ \lim_{x \to a} f(x) = f(a) \]
And since \( g \) is continuous at \( x = a \), we have:
\[ \lim_{x \to a} g(x) = g(a) \]

### Step 3: Consider the Function \( f - g \)
Now, we need to find the limit of \( f - g \) as \( x \) approaches \( a \):
\[ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \]

### Step 4: Substitute the Limits
Using the continuity of \( f \) and \( g \):
\[ \lim_{x \to a} (f(x) - g(x)) = f(a) - g(a) \]

### Step 5: Conclusion
Thus, we can conclude that:
\[ \lim_{x \to a} (f(x) - g(x)) = f(a) - g(a) = (f - g)(a) \]
This shows that \( f - g \) is continuous at \( x = a \).

### Final Statement
Therefore, if \( f \) and \( g \) are both continuous at \( x = a \), then \( f - g \) is also continuous at \( x = a \).

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