Let $f$, be a continuous function in $[0,1]$, then $\lim_{n \rightarrow \infty} \sum_{j=0}^n \frac{1}{n} f\left(\frac{j}{n}\right)$ is

• A

  $\frac{1}{2} \int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
• B

  $\int_{\frac{1}{2}}^{1} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
• Correct answer
  C

  $\int_{0}^{1} f(x) d x$
• D

  $\int_{0}^{\frac{1}{2}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$

Solution

\(\lim _{n \rightarrow \infty} \sum_{i=0}^{n} \frac{1}{n} f\left(\frac{i}{n}\right)\) Let \(1 / n \rightarrow d x\)

Hence, the correct answer is option: 'C'.