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Differentiability using limit definition
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[QUOTE="brmath, post: 4532340, member: 486151"] You are correct that to make this differentiable you have to pick k = 2. And you are correct in your reasoning about how you got there. The discussion about continuity is not relevant. You weren't asked to make it continuous or show that it is continuous. You are writing down correct facts about continuity, but perhaps a little context is needed? For a function to be differentiable is a stronger condition than for it to be continuous. Lots of functions like f(x) = |x| are continuous but not differentiable at one or several points. Someone has even defined a horror which is continuous everywhere and differentiable nowhere. So continuous cannot imply differentiable. But differentiability is about smoothness. If lim##_{x \rightarrow x_0} \frac {f(x) - f(x_0)}{x-x_0}## is to exist, how could f be discontinuous at ##x_0##? Either the left and right limits won't be the same, or ##f(x) - f(x_0)## doesn't go to zero. (Can you see why?). Note also that no one is saying f has to be continuous anywhere near ##x_0##. It could have all sorts of discontinuities nearby. But if it's differentiable at ##x_0## it is continuous at that point. [/QUOTE]
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Differentiability using limit definition
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