If a,ba,b are linearly independent functions on an interval II, are they linearly independent on any interval JJ contained in II?

Let’s say a, b are linearly independent functions on an interval I. Are they linearly independent on any interval J contained in I? If so, how do I prove it?

Let’s say a, b are instead linearly dependent functions on an interval I. Are they linearly dependent on any interval J contained in I? If so, how do I prove it?

I have a feeling I’m supposed to use the Wronskian determinant for these but I’m not sure how to apply it.

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What happens for a=xa= x, b=x1(−∞,1]+x21(1,∞)b= x \mathbf{1}_{(-\infty,1]}+ x^2 \mathbf{1}_{(1,\infty)} ?
– i707107
Oct 20 at 23:53

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No, don’t use the Wronskian determinant! Use your brain. Look for counterexamples. As a small hint, the answer to one of those questions is Yes, and the answer to the other one is No.
– TonyK
Oct 21 at 0:12

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