Pointwise estimate for a sequence of mollified functions

Question

In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote

... let $\eta_\delta$ be the standard mollifier sequence. Let $u_\delta=\eta_\delta\star u$ and note that for any $c\in (a,b)$ $$|u_\delta(x)-u_\epsilon(x)|\le \int_c^x |u'_\delta (t)-u'_\epsilon(t)|dt+|u_\delta (c)-u_\epsilon(c)|\tag{1}.$$

Since I am new to this subject, I'd like to know which theorem/lemma Tomás used to get inequality (1).

I am unable to, my reputation points have to be over 50 to comment under the answer. — , Aug 28 '14 at 10:47
@barakmanos Spurious upvotes are to be avoided. This issue was discussed, and resolved, on Meta. — , Aug 28 '14 at 14:46
@Thursday: OK, sorry, I cannot revert that by now (BTW, it looks as though I wasn't the only one)... — barak manos, Aug 28 '14 at 14:47

score 3 · Accepted Answer · answered Aug 28 '14 at 14:42

3

By the fundamental theorem of calculus, we have that $$u_\delta(x)=u_\delta (c)+\int_c^xu_\delta'(t)dt,$$

Can you conclude now?

answered Aug 28 '14 at 14:42

Tomás

22,985

Indeed I can. Thank you very much for your clarification. Sorry for causing such a fuss due to my lack of reputation points. – Aug 28 '14 at 15:20

Pointwise estimate for a sequence of mollified functions

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