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**Theorem**:

**Every continuous function f defined on [a, b] is Riemann integrable**

**Proof**:

Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b],

â€‹\( \therefore \)â€‹Â By definition of uniformly continuous function,

â€‹\( \forall \epsilon>0, \exists \delta>0 \)â€‹ (â€‹\( \delta \)â€‹ depends only on â€‹\( \epsilon \)â€‹) such that,

\[ \|x-y\|<\delta\implies\|f(x)-f(y)\|<\frac{\epsilon}{b-a},\forall x, y\in[a, b] \]

â€‹ ……………. (1)

Let â€‹\( P=\{a=x_0, x_1, … ,x_n=b\} \)â€‹ be a partition of [a, b] such that â€‹\( \|P\|<\delta \)â€‹.

Since, f is continuous on [a, b], it is continuous on â€‹\( [x_{k-1}, x_k] \)â€‹.

Also, f is continuous on closed and bounded interval then f attains its bounds.

Let â€‹\( y_k, z_k\in [x_{k-1}, x_k] \)â€‹Â such that,

â€‹\( m_k=f(y_k) \)â€‹ and â€‹\( M_k=f(z_k) \)â€‹, k=1,2, … , nÂ

â€‹\( \|y_k-z_k\|=\leq\|x_k-x_{k-1}\|\leq\|P\|<\delta \)â€‹

â€‹\( \|f(y_k)-f(z_k)\|<\frac{\epsilon}{b-a} \)â€‹Â ….. from (1)

i.e. â€‹\( \|m_k-M_k\|<\frac{\epsilon}{b-a} \)â€‹

i.e. â€‹\( \|M_k-m_k\|<\frac{\epsilon}{b-a} \)â€‹

â€‹\( \therefore M_k-m_k<\frac{\epsilon}{b-a} \)â€‹Â (â€‹\( \because M_k\geq m_k \)â€‹) ….. (2)

Consider,

U(f, P)-L(f, P)Â

â€‹\( =\displaystyle\sum_{k=1}^{n}M_k.\Delta_{x_k}-\displaystyle\sum_{k=1}^{n}m_k.\Delta_{x_k} \)â€‹

â€‹\( =\displaystyle\sum_{k=1}^{n}(M_k-m_k)\Delta_{x_k} \)â€‹

â€‹\( <\displaystyle\sum_{k=1}^{n}\Big(\frac{\epsilon}{b-a}\Big)(x_k-x_{k-1}) \)â€‹

â€‹\( =\Big(\frac{\epsilon}{b-a}\Big)\displaystyle\sum_{k=1}^{n}(x_k-x_{k-1}) \)â€‹

â€‹\( =\frac{\epsilon}{(b-a)}(b-a) \)â€‹

â€‹\( =\epsilon \)â€‹

â€‹\( \therefore U(f, P)-L(f, P)<\epsilon \)â€‹

â€‹\( \therefore \)â€‹ for â€‹\( \epsilon>0 \)â€‹, â€‹\( \exists \)â€‹ a partition P such that,

â€‹\( U(f, P)-L(f, P)<\epsilon \)â€‹Â

â€‹\( \therefore \)â€‹Â by Riemann criterion,

f is Riemann integrable.

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