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