# Pointwise estimate for a sequence of mollified functions

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).

• Post it as a comment under his answer. – barak manos Aug 28 '14 at 10:18
• I am unable to, my reputation points have to be over 50 to comment under the answer. – user156182 Aug 28 '14 at 10:47
• OK, 6 more up-votes and you're there. Here's one from me... – barak manos Aug 28 '14 at 10:48
• @barakmanos Spurious upvotes are to be avoided. This issue was discussed, and resolved, on Meta. – user147263 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

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