Let "f" be a continuous function on the closed interval "[0, 1]" with range also contained in "[0, 1]". Prove that that there exists a value "x in [0, 1]" such that "f(x)=x." Is the claim still valid if "f" is not continuous?

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Let &#34;f&#34; be a continuous function on the closed interval &#34;[0, 1]&#34; with range also contained in &#34;[0, 1]&#34;.  Prove that that there exists a value  &#34;x in [0, 1]&#34; such that &#34;f(x)=x.&#34;  Is the claim still valid if &#34;f&#34; is not continuous?

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