If there exist a first derivative of a function at any point then the funtion is continuous at that point.
What if the second derivative of that function is also exist at that point ? Does this mean that function is smooth at that point ? Means instead of taking sharp bend the function is having curved shape ?
If you found the first derivative then it is enough to say that the function is continuous. If you have also found that second derivative exists then it gives you more information about your function about what rate the funtions slope is increasing or decreasing at that point.
Also if the function were sharp at a particular point then first derivative itself wouldnt have existed at that point
There is no easily visible property of a function's graph that will tell you if the second derivative exists.
For example, one of the simplest examples of a function that is once but not twice differentiable at $0$ is $x\mapsto x\cdot |x|$ -- but its graph is not really visibly less smooth than that of $x\mapsto x^3$, which is arbitrarily often differentiable -- at least not to me.
"Not taking sharp bends" is a somewhat good intuition about what you get for being once differentiable everywhere. Note that this is not what the word "smooth" most often means in mathematics, though -- here "smooth" tends to means "has higher derivatives of every order" (which, as I argue above, is not necessarily really an intuitive concept).
It's easier to talk about discontinuity than continuity.
If a function has a discontinuity (i.e. is not continuous), then it typically means that there is "jump" in the function value, so its graph has a gap in it. This is not 100% true, though. There are weird functions that oscillate infinitely fast in a small region, and these are also not continuous. But let's ignore strange functions like that, for now.
If there is a discontinuity in the first derivative of a function, it means that its graph has a sharp corner -- a place where there is an abrupt change in direction.
If there is a discontinuity in second derivative, it means there is an abrupt change in curvature (or radius of curvature). Some people can see these discontinuities, and some people can't. Anyone with a background in graphics or design will certainly see them.
Things get a lot more interesting when you make surfaces from your curves, and you look at reflections in these surfaces, as you would when looking at a car, for example. Reflection "magnifies" discontinuities. A discontinuity in curvature will be clearly visible in a reflective surface, even to the untrained eye. Even a discontinuity in third derivative will be visible to a designer. That's why people who design cars don't use cubic splines (which you are studying, sounds like), because these have only continuity of first and second derivatives.
