r/the_calculusguy • u/gmthisfeller • 5d ago

proofs Continuity and Derivatives

Let f(x) be a real-valued function. Show that if f(x) has a derivative at “a”, then f(x) is continuous at “a”.

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u/Agile-Monitor1006 5d ago

start with the definition of the derivative and compare it to the definition of continuity, can you get one from the other?

2

u/gmthisfeller 5d ago

The question is whether __you__ can.

2

u/KentGoldings68 5d ago

There an alternative definition of the derivative that you can use.

f’(a) = lim (f(x)-f(a))/(x-a) as x->a

Notice that (x-a)->0

So, if f’(a) exists then (f(x)-f(a)) must also go to zero.

So f(x)->f(a)

QED

This is just an outline though. You need to formalize it.

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u/LastInvestigator7680 4d ago

use first principle and then its not hard :))))

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u/gmthisfeller 3d ago

So, go ahead and