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Consider the function f(x) = { x e^{(1/x)+(1/x)}, x \ne 0 { 0, x = 0 Then f(x) is:

1. discontinuous at every point
2. continuous and differentiable for all real x
3. continuous for every x, but not differentiable at x = 0
4. neither continuous nor differentiable at x = 0

Correct answer: continuous for every x, but not differentiable at x = 0

Solution

The function is defined to be 0 at x = 0, and as x approaches 0 from either side, the limit of f(x) equals 0, which matches the function's value at that point, indicating continuity. However, the derivative at x = 0 does not exist because the left-hand and right-hand derivatives do not match, making it not differentiable at that point.

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