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Here is "The Pasting Lemma" from Munkres Topology 2E (p108 Theo 18.3):

Let $X = A \cup B$, where A and B are closed in X. Let $f: A \to Y \text{ and } g: B \to Y$ be continuous. If $f(x) = g(x)$ for every $x \in A \cap B$, then f and g combine to give a continuous function $h: X \to Y$, defined by setting $h(x) = f(x) \text{ if } x \in A\text{, and } h(x) = g(x)\text{ if } x \in B.$

Munkres has there an easy example with function $h: R \to R$.

But i did not come up with a more topological example. Also using google has returned no fruitful results.

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    So what exactly is what you are looking for? Cause constructing two functions which satisfy these conditions and then combining them seems to be quite a simple task. – 2017-02-09
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    i thought about a topological example with involving S^n, open disks, etc. Sure of course Munkres example is nice and simple and reminds on the analysis course, but i doesnt show, why "the pasting lemma" is relevant to topology. – 2017-02-09

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Ok, so one real usage of this lemma is in the construction of the fundamental group.

So assume that $f,g:[0,1]\to X$ are paths (i.e. simply continous functions). Now consider

$$f':\big[0,\frac{1}{2}\big]\to X$$ $$f'(t)=f(2t)$$ $$g':\big[\frac{1}{2},1\big]\to X$$ $$g'(t)=g(2t-1)$$

These two functions are continous (being a composition of continous functions). If $f(1)=g(0)$ then $f'(\frac{1}{2})=g'(\frac{1}{2})$ so $f'$ and $g'$ have common value on $\{\frac{1}{2}\}=[0,\frac{1}{2}]\cap[\frac{1}{2},1]$. So pasting lemma applies in this scenario and we can combine $f'$ and $g'$ to get a new continous path $f*g:[0,1]\to X$. This combinantion is also known as path composition.

Very important concept in the homotopy theory. The main property of this composition is that it preserves homotopies. I.e. if $f_1\sim f_2$ and $g_1\sim g_2$ are pairwise homotopic paths then $f_1*g_1\sim f_2*g_2$ (as long as it makes sense, i.e. they have compatible beginings and ends).

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    Wow, how could I overlook this connection? I always wondered why rotman introduced the gluing lemma (= the pasting lemma) in the homotopy chapter. Now i can see it. – 2017-02-09