- Thread starter Vol
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Do not be freaked out. If you study mathematics this happens frequently.Having difficulty with the concept of additivity in linearity. In order for a function to be linear it must be additive. f(x + y) = f(x) + f(y). y = mx + b is additive only if b = 0. Then why are all y = mx + b lines? Also, how are linear differential equations additive?

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Read this link.

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Vol, I can think right off hand where the adjective

As I said before: if you do any advanced mathematics you must learn to deal with that reality.

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If so, then, yes, if f satisfies both f(x+ y)= f(x)+ f(y) and f(ax)= af(x) then f(ax+ by)= f(ax)+ f(by) (by additivity) and then = af(x)+ bf(y) (by homogeneity). Conversely if f satisfies f(ax+ by)= af(x)+ bf(y) then f(x+ y)= f(x)+ f(y) (take a= b= 1) and f(ax)= af(x) (take b= 0).