Elementary function, only in the definition domain and the definition interval must be continuous. Didn't say it would conduct electricity.
For example, the cubic of f(x)= x follows this elementary function and is continuous at x=0, but it is not differentiable.
Elementary function is a function obtained by finite four operations and compound operations of basic elementary function.
Advanced mathematics divides basic elementary functions into five categories: power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function.
Mathematical analysis divides basic elementary functions into six categories: power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function and constant function.