Complete the following sentences in terms of a limit.


a. A function is continuous from the left at a if _____.

Verified step by step guidance

1. Step 1: Understand the concept of continuity from the left. A function is continuous from the left at a point \( a \) if the left-hand limit of the function as \( x \) approaches \( a \) is equal to the function's value at \( a \).
2. Step 2: Express the left-hand limit mathematically. The left-hand limit of a function \( f(x) \) as \( x \) approaches \( a \) from the left is denoted as \( \lim_{{x \to a^-}} f(x) \).
3. Step 3: State the condition for left continuity. For the function \( f(x) \) to be continuous from the left at \( a \), the condition \( \lim_{{x \to a^-}} f(x) = f(a) \) must be satisfied.
4. Step 4: Consider the implications. This means that as \( x \) gets arbitrarily close to \( a \) from values less than \( a \), the function values \( f(x) \) should approach \( f(a) \).
5. Step 5: Summarize the sentence. A function is continuous from the left at \( a \) if \( \lim_{{x \to a^-}} f(x) = f(a) \).