Suppose that f AR is continuous at a number a E A with f(a) > 0. Show that there exist numbers L and 8 such that f(x) > L for every x EN(a; 8) nA. (Here, N(a; 6) denotes the open interval/neighborhood (a - 8, a + 8).)

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A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a&lt;x*&lt;b then…

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

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A: In this question, we will check the given function is continuous.

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A: fx=x for all real xin the open interval  0,1

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A: I have used intermidiate value theoram of continuous function.

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