FunctionContinuous

Whether f is continuous for every real x (or, given a domain restriction, throughout it).

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFunctionContinuous✓

FunctionContinuous(f, x)True iff f has no singularity anywhere on the reals.

FunctionContinuous(f, x, domain)True iff f, restricted to domain (a one-sided or bounded inequality in x), is continuous throughout it.

Typed signatures

Type	Package	Applies
`(expression, symbol, expression?) -> expression`	`analytic`	always

Domain: Elementary functions

Details

The 2-arg form is built on FunctionSingularities: True iff that set is identically False (no singularity at all), False otherwise -- confirmed against wolframscript: FunctionContinuous(1/x, x) is False (a pole exists) even though 1/x is continuous everywhere it IS defined, and FunctionContinuous(Sqrt(x), x) is likewise False (its domain isn't all of R), matching Wolfram exactly.
The 3-arg form checks the domain restriction sits entirely inside one of f's continuous intervals, closedness-aware: Sqrt's own zero is a closed, fine boundary (FunctionContinuous({Sqrt(x), x>=0}, x) is True) while Log's is closed-and-broken (FunctionContinuous({Log(x), x>=0}, x) is False, since Log(0) itself is undefined) -- both confirmed against wolframscript's own FunctionContinuous[{f, cond}, x].
Declined: Tan in the 3-arg form (an unbounded discrete exclusion set, not a finite union of intervals); a domain restriction that isn't a plain one-sided or chained inequality in x; and anything FunctionSingularities itself declines.

Examples

10 more kept as data, checked against the oracles

Basic #

$1/x$ has a pole, so it is not continuous for ALL real $x$

\mathrm{FunctionContinuous}(\frac{1}{x}, x)

Out

\mathrm{False}

Known value · Wolfram 15.0.0, hand cross-checked

a polynomial is entire

\mathrm{FunctionContinuous}(x^2, x)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

Sine is continuous on the reals

\mathrm{FunctionContinuous}(\sin(x), x)

Out

\mathrm{True}

Scope #

$1/x$ restricted to $x>0$ avoids its only pole

\mathrm{FunctionContinuous}(\frac{1}{x}, x, x\gt0)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

$\sqrt x$ is continuous on its own closed domain boundary $x=0$

\mathrm{FunctionContinuous}(\sqrt{x}, x, x≥0)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

Applications #

Polynomials are continuous

\bigl\lbrack\mathrm{FunctionContinuous}(x, x), \mathrm{FunctionContinuous}(x^2+1, x), \mathrm{FunctionContinuous}(x^3-2x^2-5x+6, x)\bigr\rbrack

Out

\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack

Sine, cosine and exp are continuous

\bigl\lbrack\mathrm{FunctionContinuous}(\sin(x), x), \mathrm{FunctionContinuous}(\cos(x), x), \mathrm{FunctionContinuous}(e^{x}, x)\bigr\rbrack

Out

\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack

About

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/function-properties.ts