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I once heard that in real analysis you should expect to draw lots of pictures. Boy were they right. My question is, is it better to draw those pictures yourself or to rely on Desmos? On the one hand, drawing it yourself (or even better, mentally picturing it yourself) is good exercise, but on the other hand mighn't it be a bit pointless? Akin to instisting on doing arithmetic instead of using a calculator? What does MathStackExchange think?

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There can be benefits to both. Sketching the graph yourself may allow you to notice some characteristics of a function that you wouldn’t otherwise just looking at the graph, because drawing it yourself forces you to pay attention to the details. On the other hand, Desmos will give a more accurate depiction of the graph, and it is easier to examine a graph more in-depth than when having to sketch by hand.

Perhaps a solution is to first graph the function yourself and then check with Desmos. This allows you to analyze different properties of the graph yourself without relying on Desmos and then to further examine the function once you understand these properties.

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By all means, use all aids you have at hand to develop intuition and test hypotheses.

Just don't use a picture as a proof, specially in a real analysis course.

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