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Omega017
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How do I go about determining where [tex] f(x,y) = \sqrt{|x| + |y|} [/tex] is differentiable?
Differentiable refers to the property of a function where it is smooth and has a well-defined tangent line at every point. In other words, a differentiable function does not have any sharp corners or breaks in its graph.
To determine differentiability at a point, we can use the definition of a derivative. If the limit of the difference quotient (change in y divided by change in x) exists as x approaches the given point, then the function is differentiable at that point.
Yes, it is possible for a function to be differentiable at some points but not others. This occurs when the function has a sharp corner, vertical tangent, or discontinuity at certain points.
Continuity refers to the property of a function where there are no breaks or gaps in its graph. A function can be continuous but not differentiable, as it may have a sharp corner or cusp at a certain point. Differentiability requires the function to be smooth and have a well-defined tangent line at every point.
No, a function cannot be differentiable but not continuous. If a function is differentiable at a point, it must also be continuous at that point. This is because differentiability implies continuity, but the reverse is not necessarily true.