Cumulative Distribution function support change

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Consider parameter $u$ has a cumulative distribution function $F$, with continous density $f$, and that has support $[m,n]$. Suppose the lower support point $m$ is changed to a new point $v$, where $v>m$. In this case, can I describe the new CDF function with the old $F$ or I need to develop a new PDF and therefore CDF functions?