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Prove that the greatest integer function...

Prove that the greatest integer function defined by `f(x) = [x], 0 < x < 3` is not differentiable at `x = 1 and x = 2`.

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To prove that the greatest integer function defined by \( f(x) = \lfloor x \rfloor \) for \( 0 < x < 3 \) is not differentiable at \( x = 1 \) and \( x = 2 \), we will analyze the left-hand and right-hand derivatives at these points. ### Step 1: Check differentiability at \( x = 1 \) 1. **Calculate the left-hand derivative at \( x = 1 \)**: \[ f'(1^-) = \lim_{h \to 0} \frac{f(1) - f(1 - h)}{h} \] ...
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