2. Prove or disprove this statement: if f;g: R !R are continuous, then their product fgis continuous. 3. Prove or disprove this statement: if f;g: R !R are uniformly continuous, then their product fgis uniformly continuous. 4. Give a **function** **f**: [0;1] !R that is not Riemann **integrable**, **and** prove that it is not. 1. **Suppose** **f** (x) is **integrable** over the interval [a, b] and m and M are minimal and maximal value of the **function** , that is m < **f** (x) < M for all x in [a, b], then Geometric meaning of the above inequality is that the area under the graph of **f** ( x ) over the interval [ a , b ] is contained inside the rectangles with the same base ( b - a ) and of .... This is also a special case of something more general: if L p is the space of all p-**integrable functions** (**f** such that ∫**f** p <∞), then the dual space (L p) * of linear functionals on L p is naturally isomorphic to L q, where q satisfies 1/p + 1/q = 1.The isomorphism sends **g** in L q to the **functional f**∈L p--> ∫fg, and the fact that the latter integral is finite is Hölder's inequality.