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Can someone give me an example of two random variables with the same probability distribution but two different density functions? I understand we can do this by changing the density function on a point. Will the value of the distribution function not change due to this. If we make the change does this become a mixed distribution?

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There will be no change. Integrals are insensitive to changes on a "small enough" set. –  André Nicolas Dec 18 '12 at 7:23
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Try two densities of the same distribution, for example $f=\mathbf 1_{[0,1]}$ and $g=f\cdot\mathbf 1_{\mathbb R\setminus\mathbb Q}$.

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