Intuition of Limits and Continuity
Discontinuity
The easiest way to understand continuity is to understand discontinuity. There are four types of discontinuities - removable, jump, infinite, and oscillating - which we will showcase here.
The function below shows a removeable discontinuity at $x = 3$.
The function is equal to the blue curve, except at $x = 3$. Instead of an equation equal to $2$ as the blue curve would suggest, it instead is equal to $2$. The function is also sometimes said to contain a hole at $x = 3$.
The function below is said to contain an finite/jump discontinuity at $x = 3$.
In this example, the function stops at the point $(3, 1)$ and starts back up again at a different point $(3, 3)$.
The function below is said to contain an infinite discontinuity at $x = 3$.
In this example, there is a vertical asymptote at $x = 3$, meaning the function tends to infinity around $x = 3$.
The function below is said to contain an oscillating discontinuity at $x = 3$.
In this example, the value of $f(x)$ is undefined at $x = 3$ because the function oscillates between the $y$-values $0$ and $4$. As the function gets closer to the point $x = 3$, it oscillates more and more, tending towards infinity. Thus, there is no definable value for the function to take at $x = 3$.
Intuition of Limits
Consider the removeable discontinuity example, denote the function $f(x)$. Even though by definition $f(x) = 3$, there is a sense that saying $f(x) = 2$ is more useful information. If you consider the point $f(x + \epsilon)$ for a very small $\epsilon$, you would be much closer to the point $(3, 2)$ than $(3, 3)$. In other words, if we approach $x = 3$ along the curve of the function, we will approach the point $(3, 2)$.
Consider the jump discontinuity example. There is actually no definable limit in this situation. From the perspective of the left curve, the limit seems to tend towards $1$ as $x$ approaches $3$. However, from the perspective of the right curve, the limit seems to tend towards $3$ as $x$ approaches $3$. Thus, the total limit does not exist. However, this should give an intuition for the concept of a left-handed and right-handed limit.
The infinite discontinuity occurs due to limits that tend towards infinity, which I am going to cover at the end of the series. The oscillating discontinuity is actually pretty complex. There is a really good video analyzing this function here.
Definition of Finite Limits
Let $f$ be any function, $a \in \mathbb{R}$ not necessarily in the domain of $f$, and $L \in \mathbb{R}$ not necessarily in the range of $f$.
The notation $\displaystyle \lim_{x \rightarrow a^-} f(x) = L$ defines the left-hand limit, and it means
\[\forall \epsilon > 0 \quad \exists \delta > 0 \quad \text{s.t.} \quad x > a - \delta \implies \lvert f(x) - L \rvert < \epsilon\]The notation $\displaystyle \lim_{x \rightarrow a^+} f(x) = L$ defines the right-hand limit, and it means
\[\forall \epsilon > 0 \quad \exists \delta > 0 \quad \text{s.t.} \quad x < a + \delta \implies \lvert f(x) - L \rvert < \epsilon\]The notation $\displaystyle \lim_{x \rightarrow a} f(x) = L$ defines the limit, and it means
\[\forall \epsilon > 0 \quad \exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \implies \lvert f(x) - L \rvert < \epsilon\]Notice that $\displaystyle \lim_{x \rightarrow a} f(x) = L$ if and only if $\displaystyle \lim_{x \rightarrow a^-} f(x) = L$ and $\displaystyle \lim_{x \rightarrow a^+} f(x) = L$
If such an $L$ satisfies the above, then the limit exists and is equal to $L$, otherwise, the limit does not exist and equality is left undefined.
Definition of Continuity
We say that $f$ is continuous at $a$ if $\displaystyle \lim_{x \rightarrow a} f(x) = f(a)$. Written using the limit definition,
\[\forall \epsilon > 0 \quad \exists \delta > 0 \quad \text{s.t.} \quad 0 < \lvert x - a \rvert < \delta \implies \lvert f(x) - f(a) \rvert < \epsilon\]We say that $f$ is continuous if it is continuous for all $a$ in the domain of $f$.
Intuitively, a function is continuous if you can draw it without picking up your pencil. This is what the definition is essentially saying. If the limit is always equal to the value of the function, then locally the function always equals what it is approaching.