I have a piece-wise function defined on the real line. The pieces are connected continuously, and the second derivative of each piece is strictly positive. Does this means that the function is convex?
More generally, where can I see about properties of such function or such construction?
On
No, you function can be non-convex as a whole. Take for example $$f(x) = \begin{cases}(x-1)^2,&x\ge 0,\\(x+1)^2,&x<0.\end{cases}.$$
It has convex pieces, yet globally it is neither convex nor concave.
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Moreover, you proposed condition that the function is increasing doesn't help, either. Take $$g(x) = \begin{cases}x^2+1,&x\ge 0,\\e^{x},&x<0.\end{cases}.$$
The function $f(x)=(x-1)^2$ if $x \ge 0$ and $(x+1)^2$ if $x<0$ is not convex, but the two pieces are continuously connected (at $(0,1)$) and each has strictly positive second derivative, namely $2.$