I'm not sure if $f(0)$ is a minimum of the function: $$ f(x)= \begin{cases} x^2(\sin\left(\frac 1x\right)+1) & \text{for } x\ne0, \\ 0 & \text{for }x=0 \end{cases} $$
and if this is a good example of why if $f$ is continuous on $I$ plus $x_1\in I$ and $f(x_1)$ is a minimum of $f$ then there is a right neighborhood of $x_1$ in which $f$ is increasing?