Let f be a continuous function in [0, 1], the | vedclass

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(C) The given expression is a Riemann sum for the definite integral of the function $f(x)$ over the interval $[0, 1]$.By the definition of the definit

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

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(C) The given expression is a Riemann sum for the definite integral of the function $f(x)$ over the interval $[0, 1]$.By the definition of the definite integral as the limit of a sum:$\lim_{n \rightarrow \infty} \sum_{j=0}^n \frac{1}{n} f\left(\frac{j}{n}\right) = \int_{0}^{1} f(x) dx$Here, we substitute $\frac{j}{n} = x$ and $\frac{1}{n} = dx$.The lower limit is $\lim_{n \rightarrow \infty} \frac{0}{n} = 0$.The upper limit is $\lim_{n \rightarrow \infty} \frac{n}{n} = 1$.Therefore, the limit is equal to $\int_{0}^{1} f(x) dx$.

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

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