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Multivariable Calculus Question on Integral Calculus

Which of the following functions is/are Riemann integrable on [0, 1] ?

A

f(x)=∫0x∣12−t∣dtf(x)=\int\limits^x_0|\frac{1}{2}-t|dt

B

f(x)={xsin⁡(1/x)if x≠0 0if x=0f(x)=\begin{cases} x \sin(1/x) & \text{if }x \ne0 \\\ 0 & \text{if }x=0 \end{cases}

C

f(x)={1if x∈Q∩[0,1] −1otherwisef(x)=\begin{cases} 1 & \text{if }x \in Q ∩[0,1] \\\ -1 & \text{otherwise} \end{cases}

D

f(x)={xif x∈[0,1) 0if x=1f(x)=\begin{cases} x & \text{if }x \in [0,1) \\\ 0 & \text{if } x=1\end{cases}

Answer

f(x)=∫0x∣12−t∣dtf(x)=\int\limits^x_0|\frac{1}{2}-t|dt

Explanation

Solution

The correct option is (A) : f(x)=∫0x∣12−t∣dtf(x)=\int\limits^x_0|\frac{1}{2}-t|dt, (B) : f(x)={xsin⁡(1/x)if x≠0 0if x=0f(x)=\begin{cases} x \sin(1/x) & \text{if }x \ne0 \\\ 0 & \text{if }x=0 \end{cases} and (D) : f(x)={xif x∈[0,1) 0if x=1f(x)=\begin{cases} x & \text{if }x \in [0,1) \\\ 0 & \text{if } x=1\end{cases}