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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help with the proof of Corollary 3.13 ...Corollary 3.13 reads as follows:View attachment 9520Can someone help me to prove that if \(\displaystyle f\) is continuous then \(\displaystyle f^+ = \text{max} (f, 0)\) is continuous ...My thoughts are as follows:If \(\displaystyle c\) belongs to an interval where \(\displaystyle f\) is positive then \(\displaystyle f^+\) is continuous since \(\displaystyle f\) is continuous ... further, if \(\displaystyle c\) belongs to an interval where \(\displaystyle f\) is negative then \(\displaystyle f^+\) is continuous since \(\displaystyle g(x) = 0\) is continuous ... but how do we construct a proof for those points where \(\displaystyle f(x)\) crosses the \(\displaystyle x\)-axis ... ..

Help will be much appreciated ...

Peter

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help with the proof of Corollary 3.13 ...Corollary 3.13 reads as follows:View attachment 9520Can someone help me to prove that if \(\displaystyle f\) is continuous then \(\displaystyle f^+ = \text{max} (f, 0)\) is continuous ...My thoughts are as follows:If \(\displaystyle c\) belongs to an interval where \(\displaystyle f\) is positive then \(\displaystyle f^+\) is continuous since \(\displaystyle f\) is continuous ... further, if \(\displaystyle c\) belongs to an interval where \(\displaystyle f\) is negative then \(\displaystyle f^+\) is continuous since \(\displaystyle g(x) = 0\) is continuous ... but how do we construct a proof for those points where \(\displaystyle f(x)\) crosses the \(\displaystyle x\)-axis ... ..

Help will be much appreciated ...

Peter

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