If f(x) be a continuous function and g(x) be discontinuous function then f{x)+g (x) is a discontinuous function.

Answer

To prove that \( f(x) + g(x) \) is a discontinuous function when \( f(x) \) is continuous and \( g(x) \) is discontinuous, we will analyze the definitions of continuity and discontinuity step by step.

### Step-by-Step Solution:

1. **Understanding Continuity**:
A function \( f(x) \) is continuous at a point \( x = a \) if:
- The left-hand limit \( \lim_{x \to a^-} f(x) \) exists.
- The right-hand limit \( \lim_{x \to a^+} f(x) \) exists.
- The value of the function at that point \( f(a) \) is equal to both limits:
\[ \lim_{x \to a} f(x) = f(a) \]

2. **Understanding Discontinuity**:
A function \( g(x) \) is discontinuous at a point \( x = a \) if at least one of the following conditions fails:
- The left-hand limit \( \lim_{x \to a^-} g(x) \) exists.
- The right-hand limit \( \lim_{x \to a^+} g(x) \) exists.
- The value of the function at that point \( g(a) \) is equal to both limits.

3. **Setting Up the Problem**:
Let \( f(x) \) be continuous at \( x = a \) and \( g(x) \) be discontinuous at \( x = a \). We denote:
- \( \lim_{x \to a^-} f(x) = L_f \)
- \( \lim_{x \to a^+} f(x) = L_f \)
- \( f(a) = L_f \)

For \( g(x) \):
- Let \( \lim_{x \to a^-} g(x) = L_{g^-} \)
- Let \( \lim_{x \to a^+} g(x) = L_{g^+} \)
- \( g(a) \) may not equal either of these limits.

4. **Calculating Limits of the Sum**:
We now consider the sum \( f(x) + g(x) \):
- The left-hand limit of the sum as \( x \to a \):
\[ \lim_{x \to a^-} (f(x) + g(x)) = \lim_{x \to a^-} f(x) + \lim_{x \to a^-} g(x) = L_f + L_{g^-} \]
- The right-hand limit of the sum as \( x \to a \):
\[ \lim_{x \to a^+} (f(x) + g(x)) = \lim_{x \to a^+} f(x) + \lim_{x \to a^+} g(x) = L_f + L_{g^+} \]

5. **Analyzing Continuity of the Sum**:
For \( f(x) + g(x) \) to be continuous at \( x = a \), we need:
\[ \lim_{x \to a^-} (f(x) + g(x)) = \lim_{x \to a^+} (f(x) + g(x)) = (f(a) + g(a)) \]
This implies:
\[ L_f + L_{g^-} = L_f + L_{g^+} = f(a) + g(a) \]

6. **Conclusion**:
Since \( g(x) \) is discontinuous, \( L_{g^-} \neq L_{g^+} \). Therefore, \( L_f + L_{g^-} \neq L_f + L_{g^+} \). This means:
\[ \lim_{x \to a^-} (f(x) + g(x)) \neq \lim_{x \to a^+} (f(x) + g(x)) \]
Hence, \( f(x) + g(x) \) is discontinuous at \( x = a \).