Determine if the graph of the function $f\(\left\)(x\(\right\))$is continuous and/or differentiable at $x=1$.

• A. Continuous and non-differentiable
• B. Continuous and differentiable
• C. Discontinuous and non-differentiable
• D. Discontinuous and differentiable

Answer: (A)

Verified step by step guidance

1. Step 1: To determine if the function f(x) is continuous at x=1, check if the limit of f(x) as x approaches 1 from both sides is equal to f(1).
2. Step 2: Observe the graph at x=1. The function appears to have no breaks, jumps, or holes at this point, suggesting that the limit from the left (x approaches 1 from the left) and the limit from the right (x approaches 1 from the right) are equal to f(1).
3. Step 3: To determine if the function is differentiable at x=1, check if the graph has a sharp corner or cusp at this point. Differentiability requires the function to be smooth, without any sharp turns.
4. Step 4: Examine the graph at x=1. If there is a sharp corner or cusp, the function is not differentiable at this point. If the graph is smooth, then it is differentiable.
5. Step 5: Based on the observations, conclude whether the function is continuous and/or differentiable at x=1. If the function is continuous but has a sharp corner, it is continuous and non-differentiable at x=1.