To say that function F is continuous at x=c means that there is no interruption in the graph of at c.
A function f is continuous at c if the following three conditions are met.
1. The function is defined at x=c.
2. The limit of f(x) exitst at x=c.
3. The limit of f(x) exists at x=c and it is equal to f(c).
To understand continuity on a closed interval, it's good to look at a one-sided limit.
The limit from the right means that x approaches c from values greater than c (see Figure 04).
Similarly, the limit from the left means that x approaches c from values less than c (see Figure 05).
Here are examples. f(x)=x^2 and c=4.
The limit from the right at 4 (Figure 04)
The limit from the left at 4 (Figure 05)
A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b),
the function f is continuous from the right at a and continuous from the left at b.
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