Continuous from the left

Definition

A function is continuous from the left at a point $a$ if the limit of the function as $x$ approaches $a$ from the left exists and equals the function's value at $a$. Mathematically, this is expressed as $\lim_{{x \to a^-}} f(x) = f(a)$.

5 Must Know Facts For Your Next Test

1. Continuity from the left requires both the existence of $\lim_{{x \to a^-}} f(x)$ and that $\lim_{{x \to a^-}} f(x) = f(a)$.
2. If a function is continuous at point $a$, it is also continuous from the left at $a$.
3. Graphically, continuity from the left means there will be no gap or jump when approaching from values less than $a$.
4. A function can be continuous from the left but not necessarily continuous from the right or overall continuous at that point.
5. To check for continuity from the left, consider only values of $x$ that are to the left of $a$.

Review Questions

• What must be true for a function to be continuous from the left at point $a$?
• How does being continuous at a point relate to being continuous from the left at that point?
• What graphical feature indicates continuity from the left?

Related terms

Limit: The value that a function approaches as the input approaches some value.

$\lim_{{x \to a^-}} f(x)$: The limit of function $f(x)$ as $x$ approaches $a$ from values less than $a$.

Continuous Function: A function without any interruptions, jumps, or gaps in its domain.