Suppose there is a function. Can you draw without lifting your pen? Good. That function is continuous.
$$ \textnormal{For } f(x) \textnormal{ where }x \in a, $$
$$ \lim_{x → a^{-}} f(x) = \lim_{x → a^{+}} f(x) = f(a) $$
In simple terms,
Left Hand Limit = Right Hand Limit = f(a)
I remember the above because it makes solving problems of this nature easier.
But there’s something you guys will feel a little bit easier:
$$ \lim_{x → a} f(x) = f(a) $$
Note the places where a has been used very carefully. It means that, where the y-axis should be when we approach towards a (from the right or left), we are getting the y-axis there only.
If there’s an interval (a, b) and we want to find if that part is continuous or not, then
$$ \lim_{x → a^{+}} f(x) = f(a), \lim_{x → b^{-}} f(x) = f(b) $$
Read as: RHL tending towards a = f(a) and LHL tending towards b = f(b).
Is there any sharp turn in the function? If NOT, that function is differentiable.
If the function can have a derivative, EVERYWHERE, it is differentiable.
“In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.”
<aside> 💡
A function f(x) is differentiable at a point a if its derivative exists at a.
</aside>
Differentiability is nothing but calculating slopes. Don't believe me?