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Differentiable implies continuous proof
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I give a proof that a differentiable function implies we have a continuous function. It is important to note that the converse does not hold; a continuous function does not imply a function is differentiable everywhere. Consider as an example g(x) = | x - 5 | ; it is not differentiable at the point (5,0), but it is continuous.
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References:
[1] J.A. Fridy, Introductory analysis: The theory of calculus, 2nd ed., Harcourt Academic Press, San Diego, CA, 2000.
[2] Steven A. Gaal, Point set topology, Dover Publications Inc., Mineola, NY, 2009.
[3] Edward D. Gaughan, Introduction to analysis, 5th ed., American Mathematical Society, Providence, RI, 2009.
Some products I used (I am not sponsored or endorsed by these products):
Samsung Galaxy A10e
TexMaker
Microsoft Paint
#realanalysis
🧮 Cheerful Calculations! 📐
References:
[1] J.A. Fridy, Introductory analysis: The theory of calculus, 2nd ed., Harcourt Academic Press, San Diego, CA, 2000.
[2] Steven A. Gaal, Point set topology, Dover Publications Inc., Mineola, NY, 2009.
[3] Edward D. Gaughan, Introduction to analysis, 5th ed., American Mathematical Society, Providence, RI, 2009.
Some products I used (I am not sponsored or endorsed by these products):
Samsung Galaxy A10e
TexMaker
Microsoft Paint
#realanalysis
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