Using only the classical definition which is below (and not Darboux etc) show that a continuous function over a closed interval is Riemann-integrable. That is, if a function $f$ is continuous on an interval $[a,b]$, then its definite integral over $[a,b]$ exists.

The Definite Integral as a Limit of Riemann Sums :

We say that a number $I$ is the definite integral of $f$ over $[a,b]$ if I is the limit of the Riemann sums, if the following condition is satisfied:

For any $\epsilon>0$, there exists a $\delta>0$ such that for every partition $P=\{x_0,x_1,...,x_n\}$ of $[a,b]$ with the mesh $||P||:=\max\limits_{1\leq i\leq n}(x_{i}-x_{i-1})<\delta$ and any choice of $\xi_k$ in $[x_{k−1},x_k]$, we have $$\left| \sum_{k=1}^n f(\xi_k) \Delta x_k - I \ \right| \lt \epsilon$$.

Any help is greatly needed and appreciated, as it is fine if I use Darboux but am stuck using just this definition. Thanks in advance


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