What is wrong with following reasoning?

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I have a function $f(x)$ for $x\geq0$. After checking the convexity I can tell that $f(x)$ is convex for $x>c_1$ where $c_1$ is some positive constant. Now if I replace $x$ by $x+c_1+.001$ in my original function then is it ok to say that $f(x+c_1+.001)$ is always a convex function of $x$? I will be very thankful to you for your comments. Thanks in advance.