A function is said to be continuous at a point simply if it is joined at that point.
A function is said to be differentiable at a point simply if it is smoothly joined at that point.
A function is said to be continuous at a point iff ( if and only if ) there is no break in the graph of the function at that point , i.e. , if we can draw the graph of the function through that point without lifting our pen from the paper on which we draw the graph.
A function is said to be differentiable at a point if the function is continuous at that point and there is one and only non-vertical tangent to the curve at that point.