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We had an exercise in functional analysis, which states

Show that $f = \sin (\frac1x)$ is continuous

Part of the solution for case $ x \neq 0$ was:

$$ \sin \text{ is continuous } \land \frac1x \text{ is continuous} \Longrightarrow f \text{ is continuous}$$

Why is that?

Furthermore, we had also examples which state $f + g$ is continuous if $f$ and $g$ are continuous.

What kind of literature do I need to get a better read on these properties?

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  • $\begingroup$ What definition of continuity are you working with? With respect to some it is almost immediate. $\endgroup$ Commented Nov 21, 2021 at 10:04
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    $\begingroup$ These are basic properties about continuous functions; any book on real analysis should cover these, say baby Rudin $\endgroup$ Commented Nov 21, 2021 at 10:05
  • $\begingroup$ @Keen-ameteur in this very example we have been working with the continuity definition using sequences. But for this example he just concludes from $g$ and $h$ being continuous, such that $f = g \circ h$ must be continuous too. The usual operators of $+$ and $-$ seem obvious to me. But chaining functions does not feel intuitive about the continuity property $\endgroup$ Commented Nov 21, 2021 at 10:08
  • $\begingroup$ Both cases can be dealt with by double applications of $\epsilon - \delta$ approaches $\endgroup$
    – Henry
    Commented Nov 21, 2021 at 10:20
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    $\begingroup$ Well, using sequences, it's really trivial to conclude that $f = g\circ h$ is continuous. Take sequence $(x_n)$ converging to $c$. If $h$ is continuous, then what can you tell about the sequence $h(x_n)$? Now, apply the same logic to the sequence $(h(x_n))$ and continuous function $g$. Actually, it's more involved to prove that sum of two functions is continuous then it is for composition of functions. $\endgroup$
    – Ennar
    Commented Nov 21, 2021 at 10:20

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When asked what introductory book on analysis can be studied by the self-learner, my go-to answer is Ken Binmore's superb "Mathematical Analysis". Has a considerably large number of examples, easily-digested proofs, relevant and educational problems, and full answers to everything.

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Understanding Analysis Book by Stephen Abbott.

This is also a wonderful book of real analysis with beautiful explanation and some good set of exercises. And you will get a teste of metric space and topological space. It is very useful book for beginner . Hope it helps.

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