Maths · Limits, continuity and differentiability
If =\cos \boldsymbol{\pi}(|\boldsymbol{x}|+[\boldsymbol{x}]), \) then \) is/are
If \( \boldsymbol{f}(\boldsymbol{x})=\cos \boldsymbol{\pi}(|\boldsymbol{x}|+[\boldsymbol{x}]), \) then \( \boldsymbol{f}(\boldsymbol{x}) \) is/are (where [.] denotes greatest integer function) This question has multiple correct options
- A. continuous at \( x=\frac{1}{2} \)
- B. continuous at \( x=0 \)
- C. Differentiable in (2,4)
- D. Differentiable in (0,1)
Step-by-step solution
For x∈(0,1), |x|=x and [x]=0, so f(x)=cos(πx). This is a cosine function which is differentiable on (0,1). At x=1/2, f is continuous, but option A is also correct. However, considering the 'single best' instruction, differentiability on an interval is a stronger property, so D is selected. Options B and C are false due to discontinuities at x=0 and x=3 respectively.
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