If is an even function such that `lim_(h rarr 0) (f(h)-f(0))/(h)` has some fininte non-zero value, then

Options

A. f is continuous and derivable at x =0
B. f is continuous but not differentiable at x = 0
C. f may be discontinuous at x = 0
D. None of these

Explanation

To solve the problem, we start by analyzing the given conditions and applying the definitions of limits and derivatives.

### Step-by-Step Solution:

1. **Understanding the Given Limit**:
We are given that:
\[ \lim_{h \to 0} \frac{f(h) - f(0)}{h} = k \]
where \( k \) is a finite non-zero value. This expression represents the right-hand derivative of \( f \) at \( x = 0 \).

2. **Finding the Left-Hand Derivative**:
Since \( f(x) \) is an even function, we have:
\[ f(-h) = f(h) \]
Now, we can find the left-hand derivative at \( x = 0 \):
\[ f'(0^-) = \lim_{h \to 0} \frac{f(0) - f(-h)}{h} = \lim_{h \to 0} \frac{f(0) - f(h)}{h} \]
This can be rewritten as:
\[ f'(0^-) = -\lim_{h \to 0} \frac{f(h) - f(0)}{h} = -k \]

3. **Comparing the Derivatives**:
We have:
- Right-hand derivative: \( f'(0^+) = k \)
- Left-hand derivative: \( f'(0^-) = -k \)

Since \( k \) is a non-zero finite value, we see that:
\[ f'(0^+) \neq f'(0^-) \]

4. **Conclusion on Continuity and Differentiability**:
Although both derivatives are finite, they are not equal. Therefore, the function \( f(x) \) is continuous at \( x = 0 \) (since the limit exists and equals \( f(0) \)), but it is not differentiable at \( x = 0 \).

### Final Answer:
Thus, the conclusion is:
- \( f(x) \) is continuous at \( x = 0 \) but not differentiable at \( x = 0 \).