Abstract Nonsense

Crushing one theorem at a time

Differentiability Implies Continuity


Point of Post: In this post we prove the obvious fact that if a mapping f:\mathbb{R}^n\to\mathbb{R}^m is differentiable at point it is continuous at that point.

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Motivation

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We would hope that if the total derivative is as similar to the normal derivative as we’d hope that many of the nice qualities pass over. In fact, most of them do. We prove the obvious one here that if something is differentiable (doesn’t ‘blow up’) at a point that it should be continuous (not tear) at that point.

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Differentiability Implies Continuity

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We have no preliminaries, and so we jump to the theorem:

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Theorem: Let f:U\to\mathbb{R}^m where U\subseteq\mathbb{R}^n be differentiable at some point a\in U. Then, f is continuous at a.

Proof: We merely note that since f is differentiable at A we have that for h sufficiently close to \bold{0} that

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\left\|f(a+h)-f(a)-D_f(a)(h)\right\|\leqslant \|h\|

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and so using the reverse triangle inequality we see that

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\left\|f(a+h)-f(a)\right\|\leqslant \|h\|+\|D_f(a)(h)\|\leqslant \left(1+\|D_f(a)\|_\text{op}\right)\|h\|

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where \|\cdot\|_{\text{op}} is the operator norm. Taking the limit as h\to\bold{0} of both sides gives \displaystyle \lim_{h\to\bold{0}}\left(f(a+h)-f(a)\right)=\bold{0} from where the conclusion follows. \blacksquare

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References:

1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

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May 24, 2011 - Posted by | Analysis | , , , , , ,

3 Comments »

  1. […] recalling that total differentiability implies continuity we have the following […]

    Pingback by Functions of Class C^k « Abstract Nonsense | June 4, 2011 | Reply

  2. […] Let be a sequence in , then by definition with (this is enough since we know that must be continuous) Thus, we get […]

    Pingback by The Geometry of the Derivative for Real Valued Mappings (Pt. I) « Abstract Nonsense | June 9, 2011 | Reply

  3. […] we know that real differentiable functions are continuous we may also conclude […]

    Pingback by Complex Differentiable and Holmorphic Functions (Pt. II) « Abstract Nonsense | May 1, 2012 | Reply


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